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\begin{document}

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\section*{geometry of physics - basic notions of categorical algebra}

\hypertarget{BasicNotionsOfCategoricalAlgebra}{}\subsection*{{Basic notions of Categorical algebra}}\label{BasicNotionsOfCategoricalAlgebra}

We have seen that the existence of [[Cartesian products]] in a [[category]] $\mathcal{C}$ equips is with a [[functor]] of the form

\begin{displaymath}
\mathcal{C} \times \mathcal{C}
    \overset{ (-) \times (-) }{\longrightarrow}
  \mathcal{C}
\end{displaymath}
which is directly analogous to the operation of [[multiplication]] in an [[associative algebra]] or even just in a [[semigroup]] (or [[monoid]]), just ``[[vertical categorification|categorified]]'' (Example \ref{CartesianMonoidalCategory} below). This is made precise by the concept of a \emph{[[monoidal category]]} (Def. \ref{MonoidalCategory} below).

This relation between [[category theory]] and [[algebra]] leads to the fields of \emph{[[categorical algebra]]} and of \emph{[[universal algebra]]}.

Here we are mainly interested in [[monoidal categories]] as a foundations for [[enriched category theory]], to which we turn \hyperlink{EnrichedCategories}{below}.

$\,$

\hypertarget{monoidal_categories}{}\subsubsection*{{Monoidal categories}}\label{monoidal_categories}

\begin{defn}
\label{MonoidalCategory}\hypertarget{MonoidalCategory}{}
\textbf{([[monoidal category]])}

An\_[[monoidal category]]\_ is a [[category]] $\mathcal{C}$ (Def. \ref{Categories}) equipped with

\begin{enumerate}%
\item a [[functor]] (Def. \ref{Functors})

\begin{displaymath}
\otimes
     \;\colon\;
   \mathcal{C} \times \mathcal{C}
    \longrightarrow
   \mathcal{C}
\end{displaymath}
out of the [[product category]] of $\mathcal{C}$ with itself (Example \ref{ProductCategory}), called the \textbf{[[tensor product]]},


\item an [[object]]

\begin{displaymath}
1 \in Obj_{\mathcal{C}}
\end{displaymath}
called the \textbf{[[unit object]]} or \textbf{[[tensor unit]]},


\item a [[natural isomorphism]] (Def. \ref{NaturalTransformations})

\begin{displaymath}
a
    \;\colon\;
  ((-)\otimes (-)) \otimes (-)
    \overset{\simeq}{\longrightarrow}
  (-) \otimes ((-)\otimes(-))
\end{displaymath}
called the \textbf{[[associator]]},


\item a [[natural isomorphism]]

\begin{displaymath}
\ell
\;\colon\;
(1 \otimes (-))
\overset{\simeq}{\longrightarrow}
(-)
\end{displaymath}
called the \textbf{[[left unitor]]}, and a natural isomorphism

\begin{displaymath}
r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)
\end{displaymath}
called the \textbf{[[right unitor]]},



\end{enumerate}
such that the following two kinds of [[commuting diagram|diagrams commute]], for all objects involved:

\begin{enumerate}%
\item \textbf{triangle identity}:

\begin{displaymath}
\itexarray{
     & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y)
      \\
      & {}_{\rho_x \otimes 1_y}\searrow
      && \swarrow_{1_x \otimes \lambda_y}
      &
     \\
     &&
     x \otimes y
     &&
  }
\end{displaymath}

\item the \textbf{[[pentagon identity]]}:

\begin{displaymath}
\itexarray{
    && (w \otimes x) \otimes (y \otimes z)
    \\
    & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow
     &&
    \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}}
    \\
    ((w \otimes x ) \otimes y) \otimes z
     && &&
    (w \otimes (x \otimes (y \otimes z)))
    \\
    {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow
     && &&
    \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }}
    \\
    (w \otimes (x \otimes y)) \otimes z
      && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} &&
    w \otimes ( (x \otimes y) \otimes z )
  }
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{example}
\label{CartesianMonoidalCategory}\hypertarget{CartesianMonoidalCategory}{}
\textbf{([[cartesian monoidal category]])}

Let $\mathcal{C}$ be a [[category]] in which all [[finite products]] exist. Then $\mathcal{C}$ becomes a [[monoidal category]] (Def. \ref{MonoidalCategory}) by

\begin{enumerate}%
\item taking the [[tensor product]] to be the [[Cartesian product]]

\begin{displaymath}
X \otimes Y \;\coloneqq\; X \times Y
\end{displaymath}

\item taking the [[unit object]] to be the [[terminal object]] (Def. \ref{InitialObject})

\begin{displaymath}
I \;\coloneqq\; \ast
\end{displaymath}


\end{enumerate}
Monoidal categories of this form are called \emph{[[cartesian monoidal categories]]}.

\end{example}
\begin{lemma}
\label{kel1}\hypertarget{kel1}{}
\textbf{(\href{monoidal+category#Kelly}{Kelly 64})}

Let $(\mathcal{C}, \otimes, 1)$ be a [[monoidal category]], def. \ref{MonoidalCategory}. Then the left and right [[unitors]] $\ell$ and $r$ satisfy the following conditions:

\begin{enumerate}%
\item $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;


\item for all objects $x,y \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]:

\begin{displaymath}
\itexarray{
    (1 \otimes x) \otimes y  & &
    \\
    {}^\mathllap{\alpha_{1, x, y}} \downarrow
    & \searrow^\mathrlap{\ell_x \otimes id_y} &
    \\
    1 \otimes (x \otimes y)
    & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y
  }
  \,;
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    x \otimes (y \otimes 1) & &
    \\
    {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow
    & \searrow^\mathrlap{id_x \otimes r_y} &
    \\
    (x \otimes y) \otimes 1
      &
    \underset{r_{x \otimes y}}{\longrightarrow}
      &
    x \otimes y
  }
  \,;
\end{displaymath}


\end{enumerate}
\end{lemma}
For \textbf{proof} see at \emph{[[monoidal category]]} \href{monoidal+category#kel1}{this lemma} and \href{monoidal+category#kel2}{this lemma}.

\begin{remark}
\label{CoherenceForMonoidalCategories}\hypertarget{CoherenceForMonoidalCategories}{}
Just as for an [[associative algebra]] it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a \emph{[[coherence theorem for monoidal categories]]} which states that all ways of freely composing the [[unitors]] and [[associators]] in a [[monoidal category]] (def. \ref{MonoidalCategory}) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an [[associative algebra]] altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier ``freely'' means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a [[strong monoidal functor]] from a \emph{free} monoidal category. For example if in a particular monoidal category it so happens that the object $X \otimes (Y \otimes Z)$ is actually \emph{equal} to $(X \otimes Y)\otimes Z$, then the various ways of going from one expression to another using only associators \emph{and} this equality no longer need to coincide.)

\end{remark}
\begin{defn}
\label{BraidedMonoidalCategory}\hypertarget{BraidedMonoidalCategory}{}
\textbf{([[braided monoidal category]])}

A \textbf{[[braided monoidal category]]}, is a [[monoidal category]] $\mathcal{C}$ (def. \ref{MonoidalCategory}) equipped with a [[natural isomorphism]] (Def. \ref{NaturalTransformations})

\begin{equation}
\tau_{x,y} \colon x \otimes y \to y \otimes x
\label{BraidingIsomorphism}\end{equation}
called the \textbf{[[braiding]]}, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved (``hexagon identities''):

\begin{displaymath}
\itexarray{
   (x \otimes y) \otimes z
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{\tau_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{\tau_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes \tau_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
   x \otimes (y \otimes z)
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{\tau_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes \tau_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{\tau_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
\end{displaymath}
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

\end{defn}
\begin{defn}
\label{SymmetricMonoidalCategory}\hypertarget{SymmetricMonoidalCategory}{}
A \textbf{[[symmetric monoidal category]]} is a [[braided monoidal category]] (def. \ref{BraidedMonoidalCategory}) for which the [[braiding]]

\begin{displaymath}
\tau_{x,y} \colon x \otimes y \to y \otimes x
\end{displaymath}
satisfies the condition:

\begin{displaymath}
\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}
\end{displaymath}
for all objects $x, y$

\end{defn}
\begin{remark}
\label{SymmetricMonoidalCategoriesCoherenceTheorem}\hypertarget{SymmetricMonoidalCategoriesCoherenceTheorem}{}
In analogy to the [[coherence theorem for monoidal categories]] (remark \ref{CoherenceForMonoidalCategories}) there is a [[coherence theorem for symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}), saying that every diagram built freely (see remark \ref{SymmetricMonoidalCategoriesCoherenceTheorem}) from [[associators]], [[unitors]] and [[braidings]] such that both sides of the diagram correspond to the same [[permutation]] of objects, coincide.

\end{remark}
\begin{defn}
\label{ClosedMonoidalCategory}\hypertarget{ClosedMonoidalCategory}{}
\textbf{([[symmetric monoidal category|symmetric]] [[closed monoidal category]])}

Given a [[symmetric monoidal category]] $\mathcal{C}$ with [[tensor product]] $\otimes$ (def. \ref{SymmetricMonoidalCategory}) it is called a \emph{[[closed monoidal category]]} if for each $Y \in \mathcal{C}$ the [[functor]] $Y \otimes(-)\simeq (-)\otimes Y$ has a [[right adjoint]], denoted $hom(Y,-)$

\begin{equation}
\mathcal{C}
    \underoverset
      {\underset{ [Y,-]}{\longrightarrow}}
      {\overset{(-) \otimes Y}{\longleftarrow}}
      {\bot}
  \mathcal{C}
  \,,
\label{InternalHomAdjunction}\end{equation}
hence if there are [[natural bijections]]

\begin{displaymath}
Hom_{\mathcal{C}}(X \otimes Y, Z)
   \;\simeq\;
  Hom_{\mathcal{C}}{C}(X, [Y,Z])
\end{displaymath}
for all objects $X,Z \in \mathcal{C}$.

Since for the case that $X = 1$ is the [[tensor unit]] of $\mathcal{C}$ this means that

\begin{displaymath}
Hom_{\mathcal{C}}(1, [Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z)
  \,,
\end{displaymath}
the object $[Y,Z] \in \mathcal{C}$ is an enhancement of the ordinary [[hom-set]] $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the \textbf{[[internal hom]]} between $Y$ and $Z$.

The [[adjunction counit]] (Def. \ref{AdjunctionUnitFromHomIsomorphism}) in this case is called the \emph{[[evaluation]] morphism}

\begin{equation}
X \otimes [X,Y] \overset{ev}{\longrightarrow} Y
\label{EvaluationMorphism}\end{equation}
\end{defn}
\begin{example}
\label{SetIsCartesianClosed}\hypertarget{SetIsCartesianClosed}{}
\textbf{([[Set]] is a [[cartesian closed category]])}

The [[category]] [[Set]] of all [[sets]] (Example \ref{CategoryOfSets}) equipped with its [[cartesian monoidal category]]-structure (Example \ref{CartesianMonoidalCategory}) is a [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}), hence a \emph{[[cartesian closed category]]}. The [[Cartesian product]] is the original [[Cartesian product]] of sets, and the [[internal hom]] is the [[function set]] $[X,Y]$ of functions from $X$ to $Y$

\end{example}
\begin{example}
\label{ExampleAbelianGroupsOfMonoidalCategory}\hypertarget{ExampleAbelianGroupsOfMonoidalCategory}{}
\textbf{([[tensor product]] of [[abelian groups]] is [[closed monoidal category]] [[symmetric monoidal category|symmetric]] [[monoidal category]]-[[structure]])}

The category [[Ab]] of [[abelian groups]] (as in Example \ref{ExamplesOfConcreteCategories}) becomes a [[symmetric monoidal category]] (Def. \ref{SymmetricMonoidalCategory}) with [[tensor product]] the actual [[tensor product of abelian groups]] $\otimes_{\mathbb{Z}}$ and with [[tensor unit]] the additive group $\mathbb{Z}$ of [[integers]]. Again the [[associator]], [[unitor]] and [[braiding]] isomorphism are the evident ones coming from the underlying sets.

This is a [[closed monoidal category]] with [[internal hom]] $hom(A,B)$ being the set of [[homomorphisms]] $Hom_{Ab}(A,B)$ equipped with the pointwise group structure for $\phi_1, \phi_2 \in Hom_{Ab}(A,B)$ then $(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B$.

This is the archetypical case that motivates the notation ``$\otimes$'' for the pairing operation in a [[monoidal category]].

\end{example}
\begin{example}
\label{GrpdIsACartesianClosedCategory}\hypertarget{GrpdIsACartesianClosedCategory}{}
\textbf{([[Cat]] and [[Grpd]] are [[cartesian closed categories]])}

The [[category]] [[Cat]] (Example \ref{CategoriesOfSmallCategories}) of all [[small categories]] (Example \ref{SmallCategory}) is a [[cartesian monoidal category]]-structure (Example \ref{CartesianMonoidalCategory}) with [[Cartesian product]] given by forming [[product categories]] (Example \ref{ProductCategory}).

Inside this, the [[full subcategory]] (Example \ref{FullSubcategoryOnClassOfObjects}) [[Grpd]] (Example \ref{CategoriesOfSmallCategories}) of all [[small groupoid|small]] [[groupoids]] (Example \ref{Groupoid}) is itself a [[cartesian monoidal category]]-structure (Example \ref{CartesianMonoidalCategory}) with [[Cartesian product]] given by forming [[product categories]] (Example \ref{ProductCategory}).

In both cases this yields a [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}), hence a [[cartesian closed category]]: the [[internal hom]] is given by the [[functor category]] construction (Example \ref{FunctorCategory}).

\end{example}
\begin{example}
\label{CartesianClosedMonoidalnessOfCategoriesOfPresheaves}\hypertarget{CartesianClosedMonoidalnessOfCategoriesOfPresheaves}{}
\textbf{([[categories of presheaves are cartesian closed]])}

Let $\mathcal{C}$ be a [[category]] and write $[\mathcal{C}^{op}, Set]$ for its [[category of presheaves]] (Example \ref{CategoryOfPresheaves}).

This is

\begin{enumerate}%
\item a [[cartesian monoidal category]] (Example \ref{CartesianMonoidalCategory}), whose [[Cartesian product]] is given objectwise in $\mathcal{C}$ by the [[Cartesian product]] in [[Set]]:

for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$, their [[Cartesian product]] $\mathbf{X} \times \mathbf{Y}$ exists and is given by

\begin{displaymath}
\mathbf{X} \times \mathbf{Y}
  \;\;\colon\;\;\phantom{A}
  \itexarray{
    c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1)
    \\
    {}^{\mathllap{f}}\big\downarrow
      &&
    \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } }
    \\
    c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2)
  }
\end{displaymath}

\item a [[cartesian closed category]] (Def. \ref{ClosedMonoidalCategory}), whose [[internal hom]] is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

\begin{displaymath}
[\mathbf{X}, \mathbf{Y}]
  \;\;\colon\;\;\phantom{A}
  \itexarray{
    c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} )
    \\
    {}^{ \mathllap{ f } }\big\downarrow
      &&
    \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } }
    \\
    c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} )
  }
\end{displaymath}
Here $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ denotes the [[Yoneda embedding]] and $Hom_{[\mathcal{C}^{op}, Set]}(-,-)$ is the [[hom-functor]] on the [[category of presheaves]].



\end{enumerate}
\end{example}
\begin{proof}
The first statement is a special case of the general fact that [[limits of presheaves are computed objectwise]] (Example \ref{LimitsOfPresheavesAreComputedObjectwise}).

For the second statement, first assume that $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the adjunction hom-isomorphism \eqref{HomIsomorphismForAdjointFunctors} we have for any other presheaf $\mathbf{Z}$ a [[natural isomorphism]] of the form

\begin{equation}
Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}])
    \;\simeq\;
  Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y})
  \,.
\label{InternalHomIsoInPresheaves}\end{equation}
This holds in particular for $\mathbf{Z} = y(c)$ a [[representable presheaf]] (Example \ref{RepresentablePresheaves}) and so the [[Yoneda lemma]] (Prop. \ref{YonedaLemma}) implies that if it exists, then $[\mathbf{X}, \mathbf{Y}]$ must have the claimed form:

\begin{displaymath}
\begin{aligned}
    [\mathbf{X}, \mathbf{Y}](c)
    & \simeq
    Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] )
    \\
    & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} )
    \,.
  \end{aligned}
\end{displaymath}
Hence it remains to show that this formula does make \eqref{InternalHomIsoInPresheaves} hold generally.

For this we use the equivalent characterization of [[adjoint functors]] from Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor}, in terms of the [[adjunction counit]] providing a system of [[universal arrows]] (Def. \ref{UniversalArrow}).

Define a would-be [[adjunction counit]], hence a would-be \emph{[[evaluation]]} morphism \eqref{EvaluationMorphism}, by

\begin{displaymath}
\itexarray{
    \mathbf{X} \times [\mathbf{X} , \mathbf{Y}]
    &\overset{ev}{\longrightarrow}&
    \mathbf{Y}
    \\
    \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y})
    &\overset{ev_c}{\longrightarrow}&
    \mathbf{Y}(c)
    \\
    (x, \phi) &\mapsto& \phi_c( id_c, x )
  }
\end{displaymath}
Then it remains to show that for every morphism of presheaves of the form $\mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y}$ there is a \emph{unique} morphism $\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]$ such that

\begin{equation}
\itexarray{
    \mathbf{X} \times \mathbf{A}
    && \overset{ \mathbf{X} \times \widetilde f  }{\longrightarrow} &&
    \mathbf{X} \times [\mathbf{X}, \mathbf{Y}]
    \\
    & {}_{\mathllap{ \mathrlap{f} }}\searrow
      &&
    \swarrow_{ \mathrlap{  ev } }
    \\
    && \mathbf{Y}
  }
\label{UniversalArrowConditionForEvaluationMapInPresheaves}\end{equation}
The [[commuting diagram|commutativity]] of this diagram means in components at $c \in \mathcal{C}$ that, that for all $x \in \mathbf{X}(c)$ and $a \in \mathbf{A}(c)$ we have

\begin{displaymath}
\begin{aligned}
    ev_c( x, \widetilde f_c(a) )
    & \coloneqq
    (\widetilde f_c(a))_c( id_c, x )
    \\
    & = f_c( x, a )
  \end{aligned}
\end{displaymath}
Hence this fixes the component $\widetilde f_c(a)_c$ when its first argument is the [[identity morphism]] $id_c$. But let $g \;\colon\; d \to c$ be any morphism and chase $(id_c, x )$ through the naturality diagram for $\widetilde f_c(a)$:

\begin{displaymath}
\itexarray{
    Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c)
    &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}&
    \mathbf{Y}(c)
    \\
    {}^{\mathllap{ g^\ast }}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{ g^\ast }}
    \\
    Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d)
    &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}&
    \mathbf{Y}(d)
  }
  \phantom{AAAA}
  \itexarray{
    \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \}
    \\
    \big\downarrow && \big\downarrow
    \\
    \{ (g, g^\ast(x)) \}  &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \}
  }
\end{displaymath}
This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

\begin{equation}
(\widetilde f_c(a))_d( g, x' )
  \;=\;
  f_d( x', g^\ast(a) )
\label{ComponentFormulaForEvaluationMapInPresheaves}\end{equation}
at least on those pairs $(g,x')$ such that $x'$ is in the image of $g^\ast$.

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

\begin{displaymath}
\itexarray{
    \mathbf{A}(c)
    &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c)
    \\
    {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}}
    \\
    \mathbf{A}(d)
    &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d)
  }
\end{displaymath}
which implies that \eqref{ComponentFormulaForEvaluationMapInPresheaves} must hold generally. Hence naturality implies that \eqref{UniversalArrowConditionForEvaluationMapInPresheaves} indeed has a unique solution.

\end{proof}
$\,$

The [[internal hom]] (Def. \ref{ClosedMonoidalCategory}) turns out to share all the abstract properties of the ordinary (external) [[hom-functor]] (Def. \ref{HomFunctor}), even though this is not completely manifest from its definition. We make this explicit by the following three propositions.

\begin{prop}
\label{InternalHomBifunctor}\hypertarget{InternalHomBifunctor}{}
\textbf{([[internal hom]] [[bifunctor]])}

For $\mathcal{C}$ a [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}), there is a unique [[functor]] (Def. \ref{Functors}) out of the [[product category]] (Def. \ref{ProductCategory}) of $\mathcal{C}$ with its [[opposite category]] (Def. \ref{OppositeCategory})

\begin{displaymath}
[-,-]
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  \mathcal{C}
\end{displaymath}
such that for each $X \in \mathcal{C}$ it coincides with the [[internal hom]] $[X,-]$ \eqref{InternalHomAdjunction} as a functor in the second variable, and such that there is a [[natural isomorphism]]

\begin{displaymath}
Hom(X, [Y,Z])
  \;\simeq\;
  Hom(X \otimes Y, Z)
\end{displaymath}
which is natural not only in $X$ and $Z$, but also in $Y$.

\end{prop}
\begin{proof}
We have a natural isomorphism for each fixed $Y$, and hence in particular for fixed $Y$ and fixed $Z$ by \eqref{InternalHomAdjunction}. With this the statement follows by Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject}.

\end{proof}
In fact the 3-variable adjunction from Prop. \ref{InternalHomBifunctor} even holds internally:

\begin{prop}
\label{TensorHomAdjunctionIsoInternally}\hypertarget{TensorHomAdjunctionIsoInternally}{}
\textbf{(internal tensor/hom-adjunction)}

In a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (def. \ref{ClosedMonoidalCategory}) there are [[natural isomorphisms]]

\begin{displaymath}
[X \otimes Y, Z]
   \;\simeq\;
  [X, [Y,Z]]
\end{displaymath}
whose image under $Hom_{\mathcal{C}}(1,-)$ (see also Example \ref{ChangeOfCosmosFromVToSet} below) are the defining [[natural bijections]] of Prop. \ref{InternalHomBifunctor}.

\end{prop}
\begin{proof}
Let $A \in \mathcal{C}$ be any object. By applying the natural bijections from Prop. \ref{InternalHomBifunctor}, there are composite [[natural bijections]]

\begin{displaymath}
\begin{aligned}
    Hom_{\mathcal{C}}(A , [X \otimes Y, Z])
    & \simeq
    Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z)
    \\
    & \simeq
    Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z)
    \\
    & \simeq
    Hom_{\mathcal{C}}(A \otimes X, [Y,Z])
    \\
    & \simeq
    Hom_{\mathcal{C}}(A, [X, [Y,Z]])
  \end{aligned}
\end{displaymath}
Since this holds for all $A$, the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]] (Prop. \ref{YonedaEmbedding}) says that there is an isomorphism $[ X\otimes Y, Z ] \simeq [X, [Y,Z]]$. Moreover, by taking $A = 1$ in the above and using the left [[unitor]] isomorphisms $A \otimes (X \otimes Y) \simeq X \otimes Y$ and $A\otimes X \simeq X$ we get a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    Hom_{\mathcal{C}}(1, [X\otimes Y, Z ))
      &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(1, [X, [Y,Z]])
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    Hom_{\mathcal{C}}(X \otimes Y, Z)
     &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(X, [Y,Z])
  }
  \,.
\end{displaymath}
\end{proof}
Also the key [[hom-functor preserves limits|respect]] of the [[hom-functor]] for [[limits]] is inherited by [[internal hom]]-functors

\begin{prop}
\label{InternalHomPreservesLimits}\hypertarget{InternalHomPreservesLimits}{}
\textbf{([[internal hom]] preserves [[limits]])}

Let $\mathcal{C}$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with [[internal hom]]-[[bifunctor]] $[-,-]$ (Prop. \ref{InternalHomBifunctor}). Then this bifunctor [[preserved limit|preserves]] [[limits]] in the second variable, and sends [[colimits]] in the first variable to limits:

\begin{displaymath}
[X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]
\end{displaymath}
and

\begin{displaymath}
[\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]
\end{displaymath}
\end{prop}
\begin{proof}
For $X \in \mathcal{X}$ any object, $[X,-]$ is a [[right adjoint]] by definition, and hence preserves limits by Prop. \ref{AdjointsPreserveCoLimits}.

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a [[diagram]] in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

\begin{displaymath}
\begin{aligned}
    Hom_{\mathcal{C}}(C, \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X )
    & \simeq
    Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X )
    \\
    & \simeq
    Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X )
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( (C \otimes Y(j)), X )
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( C, [Y(j), X] )
    \\
    & \simeq
    Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] )
  \end{aligned}
\end{displaymath}
which are [[natural isomorphism|natural]] in $C \in \mathcal{C}$, where we used that the ordinary [[hom-functor preserves limits]] (Prop. \ref{HomFunctorPreservesLimits}), and that the left adjoint $C \otimes (-)$ preserves colimits, since [[left adjoints preserve colimits]] (Prop. \ref{AdjointsPreserveCoLimits}).

Hence by the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]], there is an isomorphism

\begin{displaymath}
\left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right]
  \overset{\simeq}{\longrightarrow}
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X]
  \,.
\end{displaymath}
\end{proof}
$\,$

Now that we have seen [[monoidal categories]] with various extra [[properties]], we next look at [[functors]] which preserve these:

\begin{defn}
\label{LaxMonoidalFunctor}\hypertarget{LaxMonoidalFunctor}{}
\textbf{([[monoidal functors]])}

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two [[monoidal categories]] (def. \ref{MonoidalCategory}). A \emph{[[lax monoidal functor]]} between them is

\begin{enumerate}%
\item a [[functor]] (Def. \ref{Functors})

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
  \,,
\end{displaymath}

\item a [[morphism]]

\begin{equation}
\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
\label{UnitComponentOfMonoidalFunctor}\end{equation}

\item a [[natural transformation]] (Def. \ref{NaturalTransformations})

\begin{equation}
\mu_{x,y}
    \;\colon\;
  F(x) \otimes_{\mathcal{D}} F(y)
    \longrightarrow
  F(x \otimes_{\mathcal{C}} y)
\label{MonoidalComponentsOfMonoidalFunctor}\end{equation}
for all $x,y \in \mathcal{C}$



\end{enumerate}
satisfying the following conditions:

\begin{enumerate}%
\item \textbf{([[associativity]])} For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z)
      &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) )
    \\
    {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow
      &&
    \downarrow^{\mathrlap{id\otimes \mu_{y,z}}}
    \\
    F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z)
     &&
    F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) )
    \\
    {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z  }}}
    \\
    F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
      &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
    F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
  }
  \,,
\end{displaymath}
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the [[associators]] of the monoidal categories;


\item \textbf{([[unitality]])} For all $x \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]

\begin{displaymath}
\itexarray{
    1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x)
      &\overset{\epsilon \otimes id}{\longrightarrow}&
    F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}}
    \\
    F(x)
      &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}&
    F(1 \otimes_{\mathcal{C}} x  )
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{D}}  1_{\mathcal{D}}
      &\overset{id \otimes \epsilon }{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}  F(1_{\mathcal{C}})
    \\
    {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}}
    \\
    F(x)
      &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}&
    F(x \otimes_{\mathcal{C}} 1  )
  }
  \,,
\end{displaymath}
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right [[unitors]] of the two monoidal categories, respectively.



\end{enumerate}
If $\epsilon$ and alll $\mu_{x,y}$ are [[isomorphisms]], then $F$ is called a \textbf{[[strong monoidal functor]]}.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of [[braided monoidal categories]] (def. \ref{BraidedMonoidalCategory}) with [[braidings]] $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a \textbf{[[braided monoidal functor]]} if in addition the following [[commuting diagram|diagram commutes]] for all objects $x,y \in \mathcal{C}$

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{C}} F(y)
      &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}&
    F(y) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\mu_{x,y}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{y,x}}}
    \\
    F(x \otimes_{\mathcal{C}} y )
      &\underset{F(\tau^{\mathcal{C}}_{x,y}  )}{\longrightarrow}&
    F( y \otimes_{\mathcal{C}} x )
  }
  \,.
\end{displaymath}
A [[homomorphism]] $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a \textbf{[[monoidal natural transformation]]}, in that it is a [[natural transformation]] $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

compatible with the product and the unit in that the following [[commuting diagram|diagrams commute]] for all objects $x,y \in \mathcal{C}$:

\begin{displaymath}
\itexarray{
    F_1(x) \otimes_{\mathcal{D}} F_1(y)
      &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}&
    F_2(x) \otimes_{\mathcal{D}} F_2(y)
    \\
    {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow
     &&
    \downarrow^{\mathrlap{(\mu_2)_{x,y}}}
    \\
    F_1(x\otimes_{\mathcal{C}} y)
     &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}&
    F_2(x \otimes_{\mathcal{C}} y)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    && 1_{\mathcal{D}}
    \\
    & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}}
    \\
    F_1(1_{\mathcal{C}})
     &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&&
    F_2(1_{\mathcal{C}})
  }
  \,.
\end{displaymath}
We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting [[category]] of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[braided monoidal categories]], and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[symmetric monoidal categories]].

\end{defn}
\begin{remark}
\label{SymmetricMonoidalFunctor}\hypertarget{SymmetricMonoidalFunctor}{}
In the literature the term ``monoidal functor'' often refers by default to what in def. \ref{LaxMonoidalFunctor} is called a \emph{strong monoidal functor}. But for the purpose of the discussion of [[functors with smash product]] \hyperlink{FunctorsWithSmashProduct}{below}, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) then a [[braided monoidal functor]] (def. \ref{LaxMonoidalFunctor}) between them is often called a \textbf{[[symmetric monoidal functor]]}.

\end{remark}
\begin{prop}
\label{MonoidalFunctorComp}\hypertarget{MonoidalFunctorComp}{}
For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable [[lax monoidal functors]] (def. \ref{LaxMonoidalFunctor}) between [[monoidal categories]], then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms

\begin{displaymath}
\epsilon^{G\circ F}
    \;\colon\;
  1_{\mathcal{E}}
    \overset{\epsilon^G}{\longrightarrow}
  G(1_{\mathcal{D}})
    \overset{G(\epsilon^F)}{\longrightarrow}
  G(F(1_{\mathcal{C}}))
\end{displaymath}
and

\begin{displaymath}
\mu^{G \circ F}_{c_1,c_2}
  \;\colon\;
  G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2))
   \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow}
  G( F(c_1) \otimes_{\mathcal{D}} F(c_2) )
    \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow}
  G(F( c_1 \otimes_{\mathcal{C}} c_2 ))
  \,.
\end{displaymath}
\end{prop}
\hypertarget{AlgebrasAndModules}{}\subsubsection*{{Algebras and modules}}\label{AlgebrasAndModules}

\begin{defn}
\label{MonoidsInMonoidalCategory}\hypertarget{MonoidsInMonoidalCategory}{}
Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (Def. \ref{MonoidalCategory}), then a \textbf{[[monoid in a monoidal category|monoid internal to]]} $(\mathcal{C}, \otimes, 1)$ is

\begin{enumerate}%
\item an [[object]] $A \in \mathcal{C}$;


\item a morphism $e \;\colon\; 1 \longrightarrow A$ (called the \emph{[[unit]]})


\item a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the \emph{product});



\end{enumerate}
such that

\begin{enumerate}%
\item ([[associativity]]) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (A\otimes A) \otimes A
      &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
    A \otimes (A \otimes A)
      &\overset{A \otimes \mu}{\longrightarrow}&
    A \otimes A
    \\
    {}^{\mathllap{\mu \otimes A}}\downarrow
      && &&
    \downarrow^{\mathrlap{\mu}}
    \\
    A \otimes A
      &\longrightarrow&
      &\overset{\mu}{\longrightarrow}&
    A
  }
  \,,
\end{displaymath}
where $a$ is the associator isomorphism of $\mathcal{C}$;


\item ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
1 \otimes A
&\overset{e \otimes id}{\longrightarrow}&
A \otimes A
&\overset{id \otimes e}{\longleftarrow}&
A \otimes 1
\\
& {}_{\mathllap{\ell}}\searrow
& \downarrow^{\mathrlap{\mu}} &
& \swarrow_{\mathrlap{r}}
\\
&& A
}
\,,
\end{displaymath}
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.



\end{enumerate}
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}) $(\mathcal{C}, \otimes, 1, B)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a \textbf{[[commutative monoid in a symmetric monoidal category|commutative monoid in]]} $(\mathcal{C}, \otimes, 1, B)$ if in addition

\begin{itemize}%
\item (commutativity) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    A \otimes A
      && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
    A \otimes A
    \\
    & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
    \\
    && A
  }
  \,.
\end{displaymath}


\end{itemize}
A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

\begin{displaymath}
f \;\colon\; A_1 \longrightarrow A_2
\end{displaymath}
in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]

\begin{displaymath}
\itexarray{
    A_1 \otimes A_1
      &\overset{f \otimes f}{\longrightarrow}&
    A_2 \otimes A_2
    \\
    {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
    \\
    A_1 &\underset{f}{\longrightarrow}& A_2
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
    \\
    & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
    \\
    && A_2
  }
  \,.
\end{displaymath}
Write $Mon(\mathcal{C}, \otimes,1)$ for the \textbf{[[category of monoids]]} in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its [[full subcategory]] of [[commutative monoid in a symmetric monoidal category|commutative monoids]].

\end{defn}
\begin{example}
\label{MonoidGivenByTensorUnit}\hypertarget{MonoidGivenByTensorUnit}{}
Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (Def. \ref{MonoidalCategory}), the [[tensor unit]] $1$ is a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}) with product given by either the left or right [[unitor]]

\begin{displaymath}
\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1
  \,.
\end{displaymath}
By lemma \ref{kel1}, these two morphisms coincide and define an [[associativity|associative]] product with unit the identity $id \colon 1 \to 1$.

If $(\mathcal{C}, \otimes , 1)$ is a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}), then this monoid is a [[commutative monoid in a symmetric monoidal category|commutative monoid]].

\end{example}
\begin{example}
\label{TensorProductOfTwoCommutativeMonoids}\hypertarget{TensorProductOfTwoCommutativeMonoids}{}
Given a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given two [[commutative monoid in a symmetric monoidal category|commutative monoids]] $(E_i, \mu_i, e_i)$ $i \in \{1,2\}$ (def. \ref{MonoidsInMonoidalCategory}), then the [[tensor product]] $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism

\begin{displaymath}
e \;\colon\;
  1
    \overset{\simeq}{\longrightarrow}
  1 \otimes 1
    \overset{e_1 \otimes e_2}{\longrightarrow}
  E_1 \otimes E_2
\end{displaymath}
(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma \ref{kel1})) and with product morphism given by

\begin{displaymath}
E_1 \otimes E_2
   \otimes
  E_1 \otimes E_2
   \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow}
  E_1 \otimes E_1 \otimes E_2 \otimes E_2
   \overset{\mu_1 \otimes \mu_2}{\longrightarrow}
  E_1 \otimes E_2
\end{displaymath}
(where we are notationally suppressing the [[associators]] and where $\tau$ denotes the [[braiding]] of $\mathcal{C}$).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. \ref{BraidedMonoidalCategory}) and from symmetry of the braiding.

Similarly one checks that for $E_1 = E_2 = E$ then the unit maps

\begin{displaymath}
E \simeq E \otimes 1
    \overset{id \otimes e}{\longrightarrow}
  E \otimes E
\end{displaymath}
\begin{displaymath}
E \simeq 1 \otimes E
    \overset{e \otimes 1}{\longrightarrow}
  E \otimes E
\end{displaymath}
and the product map

\begin{displaymath}
\mu
    \;\colon\;
  E \otimes E
    \longrightarrow
  E
\end{displaymath}
and the braiding

\begin{displaymath}
\tau_{E,E}
    \;\colon\;
  E \otimes E
    \longrightarrow
  E \otimes E
\end{displaymath}
are monoid homomorphisms, with $E \otimes E$ equipped with the above monoid structure.

\end{example}
\begin{defn}
\label{ModulesInMonoidalCategory}\hypertarget{ModulesInMonoidalCategory}{}
Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidalCategory}), and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), then a \textbf{left [[module object]]} in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

\begin{enumerate}%
\item an [[object]] $N \in \mathcal{C}$;


\item a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the \emph{[[action]]});



\end{enumerate}
such that

\begin{enumerate}%
\item ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    1 \otimes N
      &\overset{e \otimes id}{\longrightarrow}&
    A \otimes N
    \\
    & {}_{\mathllap{\ell}}\searrow
    & \downarrow^{\mathrlap{\rho}}
    \\
    && N
  }
  \,,
\end{displaymath}
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.


\item (action property) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (A\otimes A) \otimes N
      &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}&
    A \otimes (A \otimes N)
      &\overset{A \otimes \rho}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes N}}\downarrow
      && &&
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes N
      &\longrightarrow&
      &\overset{\rho}{\longrightarrow}&
    N
  }
  \,,
\end{displaymath}


\end{enumerate}
A [[homomorphism]] of left $A$-module objects

\begin{displaymath}
(N_1, \rho_1) \longrightarrow (N_2, \rho_2)
\end{displaymath}
is a morphism

\begin{displaymath}
f\;\colon\; N_1 \longrightarrow N_2
\end{displaymath}
in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2
    \\
    {}^{\mathllap{\rho_1}}\downarrow
      &&
    \downarrow^{\mathrlap{\rho_2}}
    \\
    N_1 &\underset{f}{\longrightarrow}& N_2
  }
  \,.
\end{displaymath}
For the resulting \textbf{[[category of modules]]} of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

\begin{displaymath}
A Mod(\mathcal{C})
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{EveryObjectIsModuleOverTensorUnit}\hypertarget{EveryObjectIsModuleOverTensorUnit}{}
Given a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ (def. \ref{MonoidalCategory}) with the [[tensor unit]] $1$ regarded as a [[monoid in a monoidal category]] via example \ref{MonoidGivenByTensorUnit}, then the left [[unitor]]

\begin{displaymath}
\ell_C
    \;\colon\;
  1\otimes C \longrightarrow C
\end{displaymath}
makes every object $C \in \mathcal{C}$ into a left module, according to def. \ref{ModulesInMonoidalCategory}, over $C$. The action property holds due to lemma \ref{kel1}. This gives an [[equivalence of categories]]

\begin{displaymath}
\mathcal{C} \simeq 1 Mod(\mathcal{C})
\end{displaymath}
of $\mathcal{C}$ with the [[category of modules]] over its tensor unit.

\end{example}
\begin{example}
\label{RingsAreMonoidsInAb}\hypertarget{RingsAreMonoidsInAb}{}
The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category [[Ab]] of [[abelian groups]] from example \ref{ExampleAbelianGroupsOfMonoidalCategory}.

\begin{enumerate}%
\item A [[monoid in a monoidal category|monoid in]] $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{MonoidsInMonoidalCategory}) is equivalently a [[ring]].


\item A [[commutative monoid in a symmetric monoidal category|commutative monoid in]] in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{MonoidsInMonoidalCategory}) is equivalently a [[commutative ring]] $R$.


\item An $R$-[[module object]] in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{ModulesInMonoidalCategory}) is equivalently an $R$-[[module]];


\item The tensor product of $R$-module objects (def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}) is the standard [[tensor product of modules]].


\item The [[category of modules|category of module objects]] $R Mod(Ab)$ (def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}) is the standard [[category of modules]] $R Mod$.



\end{enumerate}
\end{example}
\begin{example}
\label{dgAlgebraAreMonoidsInChainComplexes}\hypertarget{dgAlgebraAreMonoidsInChainComplexes}{}
Closely related to the example \ref{RingsAreMonoidsInAb}, but closer to the structure we will see below for spectra, are [[monoid in a monoidal category|monoids]] in the [[category of chain complexes]] $(Ch_\bullet, \otimes, \mathbb{Z})$ from example \ref{SymmetricMonoidalCategoryOfChainComplexes}. These monoids are equivalently [[differential graded algebras]].

\end{example}
\begin{prop}
\label{MonoidModuleOverItself}\hypertarget{MonoidModuleOverItself}{}
In the situation of def. \ref{ModulesInMonoidalCategory}, the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:

\begin{displaymath}
\rho
  \;\colon\;
  A \otimes (A \otimes C)
   \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow}
  (A \otimes A) \otimes C
    \overset{\mu \otimes id}{\longrightarrow}
  A \otimes C
  \,.
\end{displaymath}
The $A$-modules of this form are called \textbf{[[free modules]]}.

The [[free functor]] $F$ constructing free $A$-modules is [[left adjoint]] to the [[forgetful functor]] $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.

\begin{displaymath}
A Mod(\mathcal{C})
    \underoverset
     {\underset{U}{\longrightarrow}}
     {\overset{F}{\longleftarrow}}
     {\bot}
  \mathcal{C}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form

\begin{displaymath}
f \;\colon\; A\otimes C \longrightarrow N
\end{displaymath}
fitting into the diagram (where we are notationally suppressing the [[associator]])

\begin{displaymath}
\itexarray{
    A \otimes A \otimes C
      &\overset{A \otimes f}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes id}}\downarrow
      &&
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes C
      &\underset{f}{\longrightarrow}&
    N
  }
  \,.
\end{displaymath}
Consider the composite

\begin{displaymath}
\tilde f
    \;\colon\;
  C
    \underoverset{\simeq}{\ell_C}{\longrightarrow}
  1 \otimes C
    \overset{e\otimes id}{\longrightarrow}
  A \otimes C
    \overset{f}{\longrightarrow}
  N
  \,,
\end{displaymath}
i.e. the restriction of $f$ to the unit ``in'' $A$. By definition, this fits into a [[commuting square]] of the form (where we are now notationally suppressing the [[associator]] and the [[unitor]])

\begin{displaymath}
\itexarray{
   A \otimes C
     &\overset{id \otimes \tilde f}{\longrightarrow}&
   A \otimes N
   \\
   {}^{\mathllap{id \otimes e \otimes id}}\downarrow
     &&
   \downarrow^{\mathrlap{=}}
   \\
   A \otimes A \otimes C
    &\underset{id \otimes f}{\longrightarrow}&
   A \otimes N
  }
  \,.
\end{displaymath}
Pasting this square onto the top of the previous one yields

\begin{displaymath}
\itexarray{
   A \otimes C
     &\overset{id \otimes \tilde f}{\longrightarrow}&
   A \otimes N
   \\
   {}^{\mathllap{id \otimes e \otimes id}}\downarrow
     &&
   \downarrow^{\mathrlap{=}}
    \\
    A \otimes A \otimes C
      &\overset{A \otimes f}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes id}}\downarrow
      &&
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes C
      &\underset{f}{\longrightarrow}&
    N
  }
  \,,
\end{displaymath}
where now the left vertical composite is the identity, by the unit law in $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation

\begin{displaymath}
f = \rho \circ (id_A \otimes \tilde f)
  \,.
\end{displaymath}
This natural bijection between $f$ and $\tilde f$ establishes the adjunction.

\end{proof}
\begin{defn}
\label{TensorProductOfModulesOverCommutativeMonoidObject}\hypertarget{TensorProductOfModulesOverCommutativeMonoidObject}{}
Given a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}, def. \ref{ClosedMonoidalCategory}), given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-[[module objects]] (def.\ref{MonoidsInMonoidalCategory}), then

\begin{enumerate}%
\item the \textbf{[[tensor product of modules]]} $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]]

\begin{displaymath}
N_1 \otimes A \otimes N_2
  \underoverset
    {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}}
    {\overset{N_1 \otimes \rho_2}{\longrightarrow}}
    {\phantom{AAAA}}
  N_1 \otimes N_1
    \overset{coeq}{\longrightarrow}
  N_1 \otimes_A N_2
\end{displaymath}
and if $A \otimes (-)$ preserves these coequalizers, then this is equipped with the left $A$-action induced from the left $A$-action on $N_1$


\item the \textbf{function module} $hom_A(N_1,N_2)$ is, if it exists, the [[equalizer]]

\begin{displaymath}
hom_A(N_1, N_2)
    \overset{equ}{\longrightarrow}
  hom(N_1, N_2)
    \underoverset
      {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}}
      {\overset{hom(\rho_1,N_2)}{\longrightarrow}}
      {\phantom{AAAAAA}}
    hom(A \otimes N_1, N_2)
  \,.
\end{displaymath}
equipped with the left $A$-action that is induced by the left $A$-action on $N_2$ via

\begin{displaymath}
\frac{
    A \otimes hom(X,N_2) \longrightarrow hom(X,N_2)
  }{
   A \otimes hom(X,N_2) \otimes X
     \overset{id \otimes ev}{\longrightarrow}
   A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2
  }
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
(e.g. \hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8})

\begin{prop}
\label{MonoidalCategoryOfModules}\hypertarget{MonoidalCategoryOfModules}{}
Given a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}, def. \ref{ClosedMonoidalCategory}), and given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}). If all [[coequalizers]] exist in $\mathcal{C}$, then the [[tensor product of modules]] $\otimes_A$ from def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} makes the [[category of modules]] $A Mod(\mathcal{C})$ into a [[symmetric monoidal category]], $(A Mod, \otimes_A, A)$ with [[tensor unit]] the object $A$ itself, regarded as an $A$-module via prop. \ref{MonoidModuleOverItself}.

If moreover all [[equalizers]] exist, then this is a [[closed monoidal category]] (def. \ref{ClosedMonoidalCategory}) with [[internal hom]] given by the function modules $hom_A$ of def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}.

\end{prop}
(e.g. \hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8})

\begin{proof}
The associators and braiding for $\otimes_{A}$ are induced directly from those of $\otimes$ and the [[universal property]] of [[coequalizers]]. That $A$ is the tensor unit for $\otimes_{A}$ follows with the same kind of argument that we give in the proof of example \ref{FreeModulesTensorProduct} below.

\end{proof}
\begin{example}
\label{FreeModulesTensorProduct}\hypertarget{FreeModulesTensorProduct}{}
For $(A,\mu,e)$ a [[monoid in a monoidal category|monoid]] (def. \ref{MonoidsInMonoidalCategory}) in a [[symmetric monoidal category]] $(\mathcal{C},\otimes, 1)$ (def. \ref{MonoidalCategory}), the [[tensor product of modules]] (def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}) of two [[free modules]] (def. \ref{MonoidModuleOverItself}) $A\otimes C_1$ and $A \otimes C_2$ always exists and is the free module over the tensor product in $\mathcal{C}$ of the two generators:

\begin{displaymath}
(A \otimes C_1) \otimes_A (A \otimes C_2)
  \simeq
  A \otimes (C_1 \otimes C_2)
  \,.
\end{displaymath}
Hence if $\mathcal{C}$ has all [[coequalizers]], so that the [[category of modules]] is a [[monoidal category]] $(A Mod, \otimes_A, A)$ (prop. \ref{MonoidalCategoryOfModules}) then the free module functor (def. \ref{MonoidModuleOverItself}) is a [[strong monoidal functor]] (def. \ref{LaxMonoidalFunctor})

\begin{displaymath}
F
    \;\colon\;
  (\mathcal{C}, \otimes, 1)
    \longrightarrow
  (A Mod, \otimes_A, A)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
It is sufficient to show that the diagram

\begin{displaymath}
A \otimes A \otimes A
   \underoverset
    {\underset{id \otimes \mu}{\longrightarrow}}
    {\overset{\mu \otimes id}{\longrightarrow}}
    {\phantom{AAAA}}
  A \otimes A
    \overset{\mu}{\longrightarrow}
  A
\end{displaymath}
is a [[coequalizer]] diagram (we are notationally suppressing the [[associators]]), hence that $A \otimes_A A \simeq A$, hence that the claim holds for $C_1 = 1$ and $C_2 = 1$.

To that end, we check the [[universal property]] of the [[coequalizer]]:

First observe that $\mu$ indeed coequalizes $id \otimes \mu$ with $\mu \otimes id$, since this is just the [[associativity]] clause in def. \ref{MonoidsInMonoidalCategory}. So for $f \colon A \otimes A \longrightarrow Q$ any other morphism with this property, we need to show that there is a unique morphism $\phi \colon A \longrightarrow Q$ which makes this [[commuting diagram|diagram commute]]:

\begin{displaymath}
\itexarray{
    A \otimes A &\overset{\mu}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}}
    \\
    Q
  }
  \,.
\end{displaymath}
We claim that

\begin{displaymath}
\phi
    \;\colon\;
  A
    \underoverset{\simeq}{r^{-1}}{\longrightarrow}
  A \otimes 1
    \overset{id \otimes e}{\longrightarrow}
  A \otimes A
    \overset{f}{\longrightarrow}
  Q
  \,,
\end{displaymath}
where the first morphism is the inverse of the right [[unitor]] of $\mathcal{C}$.

First to see that this does make the required triangle commute, consider the following pasting composite of [[commuting diagrams]]

\begin{displaymath}
\itexarray{
    A \otimes A &\overset{\mu}{\longrightarrow}& A
    \\
    {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{r^{-1}}}_{\simeq}
    \\
    A \otimes A \otimes 1
      &\overset{\mu \otimes id}{\longrightarrow}&
    A \otimes 1
    \\
    {}^{\mathllap{id \otimes e}}\downarrow
      &&
    \downarrow^{\mathrlap{id \otimes e} }
    \\
    A \otimes A \otimes A
      &\overset{\mu \otimes id}{\longrightarrow}&
    A \otimes A
    \\
    {}^{\mathllap{id \otimes \mu}}\downarrow
      &&
    \downarrow^{\mathrlap{f}}
    \\
    A \otimes A
      &\underset{f}{\longrightarrow}&
    Q
  }
  \,.
\end{displaymath}
Here the the top square is the [[natural transformation|naturality]] of the right [[unitor]], the middle square commutes by the functoriality of the tensor product $\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}$ and the definition of the [[product category]] (Example \ref{ProductCategory}), while the commutativity of the bottom square is the assumption that $f$ coequalizes $id \otimes \mu$ with $\mu \otimes id$.

Here the right vertical composite is $\phi$, while, by \hyperlink{UnitalityMonoid}{unitality} of $(A,\mu ,e)$, the left vertical composite is the identity on $A$, Hence the diagram says that $\phi \circ \mu = f$, which we needed to show.

It remains to see that $\phi$ is the unique morphism with this property for given $f$. For that let $q \colon A \to Q$ be any other morphism with $q\circ \mu = f$. Then consider the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    A \otimes 1 &\overset{\simeq}{\longleftarrow}& A
    \\
    {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq}
    & \downarrow^{\mathrlap{=}}
    \\
    A \otimes A &\overset{\mu}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}}
    \\
    Q
  }
  \,,
\end{displaymath}
where the top left triangle is the \hyperlink{UnitalityMonoid}{unitality} condition and the two isomorphisms are the right [[unitor]] and its inverse. The commutativity of this diagram says that $q = \phi$.

\end{proof}
\begin{defn}
\label{AAlgebra}\hypertarget{AAlgebra}{}
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ as in prop. \ref{MonoidalCategoryOfModules}, then a [[monoid in a monoidal category|monoid]] $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. \ref{MonoidsInMonoidalCategory}) is called an \textbf{$A$-[[associative algebra|algebra]]}.

\end{defn}
\begin{prop}
\label{AlgebrasOverAAreMonoidsUnderA}\hypertarget{AlgebrasOverAAreMonoidsUnderA}{}
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ in a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ as in prop. \ref{MonoidalCategoryOfModules}, and an $A$-algebra $(E,\mu,e)$ (def. \ref{AAlgebra}), then there is an [[equivalence of categories]]

\begin{displaymath}
A Alg_{comm}(\mathcal{C})
    \coloneqq
  CMon(A Mod)
   \simeq
  CMon(\mathcal{C})^{A/}
\end{displaymath}
between the [[category of commutative monoids]] in $A Mod$ and the [[coslice category]] of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.

\end{prop}
(e.g. \hyperlink{EKMM97}{EKMM 97, VII lemma 1.3})

\begin{proof}
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$

\begin{displaymath}
\itexarray{
    E \otimes A \otimes E
    &
    \underoverset
      {\underset{}{\longrightarrow}}
      {\overset{}{\longrightarrow}}
      {\phantom{AAA}}
    &
    E \otimes E
     &\overset{coeq}{\longrightarrow}&
    E \otimes_A E
    \\
    && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && && E
  }
  \,.
\end{displaymath}
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.

Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that

\begin{displaymath}
(\star)
  \;\;\;\;\;
  \;\;\;\;\;
  \itexarray{
    A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
\end{displaymath}
[[commuting diagram|commutes]]. Moreover it satisfies the unit property

\begin{displaymath}
\itexarray{
    A \otimes_A E
      &\overset{e_A \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && E
  }
  \,.
\end{displaymath}
By forgetting the tensor product over $A$, the latter gives

\begin{displaymath}
\itexarray{
    A \otimes E
      &\overset{e \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    \downarrow && \downarrow^{\mathrlap{}}
    \\
    A \otimes_A E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    E &=& E
  }
  \;\;\;\;\;\;\;\;
   \simeq
  \;\;\;\;\;\;\;\;
  \itexarray{
    A \otimes E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}}
    \\
    E &\underset{id}{\longrightarrow}& E
  }
  \,,
\end{displaymath}
where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square $(\star)$ above, to yield a [[commuting square]]

\begin{displaymath}
\itexarray{
    A \otimes A
     &\overset{id\otimes e_E}{\longrightarrow}&
    A \otimes E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    {}^{\mathllap{\rho}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{E}}}
    \\
    A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E
  }
  \;\;\;\;\;\;\;\;\;\;
   =
  \;\;\;\;\;\;\;\;\;\;
  \itexarray{
    A \otimes A
     &\overset{e_E \otimes e_E}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_E}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
  \,.
\end{displaymath}
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.

Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A  \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-[[module]] structure by

\begin{displaymath}
\rho
    \;\colon\;
  A \otimes E
    \overset{e_A \otimes id}{\longrightarrow}
  E \otimes E
    \overset{\mu_E}{\longrightarrow}
  E
  \,.
\end{displaymath}
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

\end{proof}
\begin{defn}
\label{LaxMonoidalFunctor}\hypertarget{LaxMonoidalFunctor}{}
\textbf{([[lax monoidal functor]])}

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two [[monoidal categories]] (def. \ref{MonoidalCategory}). A \emph{[[lax monoidal functor]]} between them is

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
  \,,
\end{displaymath}

\item a [[morphism]]

\begin{displaymath}
\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
\end{displaymath}

\item a [[natural transformation]]

\begin{displaymath}
\mu_{x,y}
    \;\colon\;
  F(x) \otimes_{\mathcal{D}} F(y)
    \longrightarrow
  F(x \otimes_{\mathcal{C}} y)
\end{displaymath}
for all $x,y \in \mathcal{C}$



\end{enumerate}
satisfying the following conditions:

\begin{enumerate}%
\item \textbf{([[associativity]])} For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z)
      &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) )
    \\
    {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow
      &&
    \downarrow^{\mathrlap{id\otimes \mu_{y,z}}}
    \\
    F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z)
     &&
    F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) )
    \\
    {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z  }}}
    \\
    F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
      &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
    F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
  }
  \,,
\end{displaymath}
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the [[associators]] of the monoidal categories;


\item \textbf{([[unitality]])} For all $x \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]

\begin{displaymath}
\itexarray{
    1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x)
      &\overset{\epsilon \otimes id}{\longrightarrow}&
    F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}}
    \\
    F(x)
      &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}&
    F(1 \otimes_{\mathcal{C}} x  )
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{D}}  1_{\mathcal{D}}
      &\overset{id \otimes \epsilon }{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}  F(1_{\mathcal{C}})
    \\
    {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}}
    \\
    F(x)
      &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}&
    F(x \otimes_{\mathcal{C}} 1  )
  }
  \,,
\end{displaymath}
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right [[unitors]] of the two monoidal categories, respectively.



\end{enumerate}
If $\epsilon$ and alll $\mu_{x,y}$ are [[isomorphisms]], then $F$ is called a \textbf{strong monoidal functor}.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of [[braided monoidal categories]] (def. \ref{BraidedMonoidalCategory}) with [[braidings]] $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a \textbf{[[braided monoidal functor]]} if in addition the following [[commuting diagram|diagram commutes]] for all objects $x,y \in \mathcal{C}$

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{C}} F(y)
      &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}&
    F(y) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\mu_{x,y}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{y,x}}}
    \\
    F(x \otimes_{\mathcal{C}} y )
      &\underset{F(\tau^{\mathcal{C}}_{x,y}  )}{\longrightarrow}&
    F( y \otimes_{\mathcal{C}} x )
  }
  \,.
\end{displaymath}
A [[homomorphism]] $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a \textbf{[[monoidal natural transformation]]}, in that it is a [[natural transformation]] $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

compatible with the product and the unit in that the following [[commuting diagram|diagrams commute]] for all objects $x,y \in \mathcal{C}$:

\begin{displaymath}
\itexarray{
    F_1(x) \otimes_{\mathcal{D}} F_1(y)
      &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}&
    F_2(x) \otimes_{\mathcal{D}} F_2(y)
    \\
    {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow
     &&
    \downarrow^{\mathrlap{(\mu_2)_{x,y}}}
    \\
    F_1(x\otimes_{\mathcal{C}} y)
     &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}&
    F_2(x \otimes_{\mathcal{C}} y)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    && 1_{\mathcal{D}}
    \\
    & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}}
    \\
    F_1(1_{\mathcal{C}})
     &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&&
    F_2(1_{\mathcal{C}})
  }
  \,.
\end{displaymath}
We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting [[category]] of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[braided monoidal categories]], and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[symmetric monoidal categories]].

\end{defn}
\begin{remark}
\label{SymmetricMonoidalFunctor}\hypertarget{SymmetricMonoidalFunctor}{}
In the literature the term ``monoidal functor'' often refers by default to what in def. \ref{LaxMonoidalFunctor} is called a \emph{strong monoidal functor}. But for the purpose of the discussion of [[functors with smash product]] \hyperlink{FunctorsWithSmashProduct}{below}, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) then a [[braided monoidal functor]] (def. \ref{LaxMonoidalFunctor}) between them is often called a \textbf{[[symmetric monoidal functor]]}.

\end{remark}
\begin{prop}
\label{MonoidalFunctorComp}\hypertarget{MonoidalFunctorComp}{}
For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable [[lax monoidal functors]] (def. \ref{LaxMonoidalFunctor}) between [[monoidal categories]], then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms

\begin{displaymath}
\epsilon^{G\circ F}
    \;\colon\;
  1_{\mathcal{E}}
    \overset{\epsilon^G}{\longrightarrow}
  G(1_{\mathcal{D}})
    \overset{G(\epsilon^F)}{\longrightarrow}
  G(F(1_{\mathcal{C}}))
\end{displaymath}
and

\begin{displaymath}
\mu^{G \circ F}_{c_1,c_2}
  \;\colon\;
  G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2))
   \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow}
  G( F(c_1) \otimes_{\mathcal{D}} F(c_2) )
    \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow}
  G(F( c_1 \otimes_{\mathcal{C}} c_2 ))
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{MonoidsPreservedByLaxMonoidalFunctor}\hypertarget{MonoidsPreservedByLaxMonoidalFunctor}{}
\textbf{([[lax monoidal functors]] preserve [[monoid in a monoidal category|monoids]])}

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})$ be two [[monoidal categories]] (def. \ref{MonoidalCategory}) and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a [[lax monoidal functor]] (def. \ref{LaxMonoidalFunctor}) between them.

Then for $(A,\mu_A,e_A)$ a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}), its image $F(A) \in \mathcal{D}$ becomes a monoid $(F(A), \mu_{F(A)}, e_{F(A)})$ by setting

\begin{displaymath}
\mu_{F(A)}
   \;\colon\;
  F(A) \otimes_{\mathcal{C}} F(A)
    \overset{}{\longrightarrow}
  F(A \otimes_{\mathcal{C}} A)
    \overset{F(\mu_A)}{\longrightarrow}
  F(A)
\end{displaymath}
(where the first morphism is the structure morphism of $F$) and setting

\begin{displaymath}
e_{F(A)}
    \;\colon\;
  1_{\mathcal{D}}
    \longrightarrow
  F(1_{\mathcal{C}})
    \overset{F(e_A)}{\longrightarrow}
  F(A)
\end{displaymath}
(where again the first morphism is the corresponding structure morphism of $F$).

This construction extends to a functor

\begin{displaymath}
Mon(F)
   \;\colon\;
  Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}})
    \longrightarrow
  Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})
\end{displaymath}
from the [[category of monoids]] of $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}) to that of $\mathcal{D}$.

Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) and $F$ is a [[braided monoidal functor]] (def. \ref{LaxMonoidalFunctor}) and $A$ is a [[commutative monoid]] (def. \ref{MonoidsInMonoidalCategory}) then so is $F(A)$, and this construction extends to a functor between categories of commutative monoids:

\begin{displaymath}
CMon(F)
   \;\colon\;
  CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}})
    \longrightarrow
  CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.

\end{proof}
$\,$

\hypertarget{EnrichedCategories}{}\subsubsection*{{Enriched categories}}\label{EnrichedCategories}

The plain definition of [[categories]] in Def. \ref{Categories} is phrased in terms of [[sets]]. Via Example \ref{CategoryOfSets} this assigns a special role to the category [[Set]] of all sets, as the ``base'' on top, or the ``[[cosmos]]'' inside which [[category theory]] takes place. For instance, the fact that [[hom-sets]] in a plain [[category]] are indeed sets, is what makes the [[hom-functor]] (Example \ref{HomFunctor}) take values in [[Set]], and this, in turn, governs the form of the all-important [[Yoneda lemma]] (Prop. \ref{YonedaLemma}) and [[Yoneda embedding]] (Prop. \ref{YonedaEmbedding}) as statements about [[presheaves]] of sets (Example \ref{CategoryOfPresheaves}).

At the same time, [[category theory]] witnesses the utility of abstracting away from concrete choices to their abstract properties that are actually used in constructions. This makes it natural to ask if one could replace the category [[Set]] by some other category $\mathcal{V}$ which could similarly serve as a ``[[cosmos]]'' inside which category theory may be developed.

Indeed, such $\mathcal{V}$-[[enriched category theory]] (see Example \ref{UnderlyingCategoryOfTopEnrichedCategory} below for the terminology) exists, beginning with the concept of $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory} below) and from there directly paralleling, hence generalizing, plain category theory, as long as one assumes the ``cosmos'' category $\mathcal{V}$ to share a minimum of abstract properties with [[Set]] (Def. \ref{Cosmos} below).

This turns out to be most useful. In fact, the perspective of [[enriched categories]] is helpful already when $\mathcal{V} =$ [[Set]], in which case it reproduces plain category theory (Example \ref{SetEnrichedCategoriesArePlainCategories} below), for instance in that it puts the [[limit|(co)limits]] of the special form of [[end|(co)ends]] (Def. \ref{EndAndCoendInTopcgSmash} below) to the forefront (discussed \hyperlink{TopologicalEndsAndCoends}{below}).

$\,$

\begin{defn}
\label{Cosmos}\hypertarget{Cosmos}{}
\textbf{([[cosmos]])}

A \emph{[[Jean Benabou|Bénabou]] [[cosmos]] for [[enriched category theory]]}, or just \emph{[[cosmos]]}, for short, is a [[symmetric monoidal category|symmetric]] (Def. \ref{SymmetricMonoidalCategory}) [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}) $\mathcal{V}$ which has all [[limits]] and [[colimits]].

\end{defn}
\begin{example}
\label{ExamplesOfCosmoi}\hypertarget{ExamplesOfCosmoi}{}
\textbf{(examples of [[cosmoi]] for [[enriched category theory]])}

The following are examples of [[cosmoi]] (Def. \ref{Cosmos}):

\begin{enumerate}%
\item $Sh(\mathcal{C})$ the [[sheaf topos]] (Def. \ref{Sheaf}) over any [[site]] (Def. \ref{Coverage}) -- by Prop. \ref{PropertiesOfSheafToposes} below.

In particular:

\begin{enumerate}%
\item [[Set]] (Def. \ref{CategoryOfSets}) equipped with its [[cartesian closed category]]-structure (Example \ref{SetIsCartesianClosed})


\item [[sSet]] $\simeq [\Delta^{op}, Set]$ (Def. \ref{sSet}, Prop. \ref{SimplicialSetsAsPresheavesOnTheSimplexCategory})



\end{enumerate}

\item [[Grpd]] (Def. \ref{CategoriesOfSmallCategories}) equipped with its [[cartesian closed category]]-structure (Example \ref{GrpdIsACartesianClosedCategory}).


\item [[Cat]] (Def. \ref{CategoriesOfSmallCategories}) equipped with its [[cartesian closed category]]-structure (Example \ref{GrpdIsACartesianClosedCategory}).



\end{enumerate}
\end{example}
\begin{example}
\label{ChangeOfCosmosFromVToSet}\hypertarget{ChangeOfCosmosFromVToSet}{}
\textbf{underlying [[set]] of an object in a [[cosmos]]}

Let $\mathcal{V}$ be a [[cosmos]] (Def. \ref{Cosmos}), with $1 \in \mathcal{V}$ its [[tensor unit]] (Def. \ref{MonoidalCategory}). Then the [[hom-functor]] (Def. \ref{HomFunctor}) out of $1$

\begin{displaymath}
Hom_{\mathcal{V}}(1,-)
  \;\colon\;
  \mathcal{V}
    \longrightarrow
  Set
\end{displaymath}
admits the [[structure]] of a [[lax monoidal functor]] (Def. \ref{LaxMonoidalFunctor}) to [[Set]], with the latter regarded with its [[cartesian monoidal category|cartesian monoidal]] [[structure]] from Example \ref{SetIsCartesianClosed}.

Given $V \in \mathcal{V}$, we call

\begin{displaymath}
Hom_{\mathcal{V}}(1,V) \;\in\; Set
\end{displaymath}
also the \emph{underlying} set of $V$.

\end{example}
\begin{proof}
Take the monoidal transformations (eq``MonoidalComponentsOfMonoidalFunctor) to be

\begin{displaymath}
\itexarray{
    Hom_{\mathcal{V}}(1, V_1)
    \times
    Hom_{\mathcal{V}}(1, V_2)
    \longrightarrow
    Hom_{\mathcal{V}}(1, V_1 \otimes V_2)
    \\
    \big( 1 \overset{f_1}{\to} V_1\;,\; 1 \overset{f_2}{\to} V_2 \big)
    &\mapsto&
    \big(
      1 \overset{\simeq}{\to} 1 \otimes 1
        \overset{f_1 \otimes f_2}{\longrightarrow}
      V_1 \otimes V_2
    \big)
  }
\end{displaymath}
and take the unit transformation \eqref{UnitComponentOfMonoidalFunctor}

\begin{displaymath}
\itexarray{
    \ast
      &\longrightarrow&
    Hom_{\mathcal{V}}(1,1)
  }
\end{displaymath}
to pick $id_1 \in Hom_{\mathcal{V}}(1,1)$.

\end{proof}
\begin{example}
\label{UnderlyingSetOfInternalHomIsHomSet}\hypertarget{UnderlyingSetOfInternalHomIsHomSet}{}
\emph{(underlying set of [[internal hom]] is [[hom-set]])*}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $X,Y \in Obj_{\mathcal{V}}$ be two [[objects]]. Then the underlying set (Def. \ref{ChangeOfCosmosFromVToSet}) of their [[internal hom]] $[X,Y] \in \mathcal{V}$ (Def. \ref{ClosedMonoidalCategory}) is the [[hom-set]] (Def. \ref{Categories}):

\begin{displaymath}
\mathcal{Hom}_{\mathcal{V}}\left( 1, [X,Y]\right)
  \;\simeq\;
  Hom_{\mathcal{V}}(X,Y)
  \,.
\end{displaymath}
This identification is the adjunction isomorphism \eqref{HomIsomorphismForAdjointFunctors} for the internal hom adjunction \eqref{InternalHomAdjunction} followed composed with a [[unitor]] (Def. \ref{MonoidalCategory}).

\end{example}
\begin{defn}
\label{TopEnrichedCategory}\hypertarget{TopEnrichedCategory}{}
\textbf{([[enriched category]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), a \textbf{$\mathcal{V}$-[[enriched category]]} $\mathcal{C}$ is:

\begin{enumerate}%
\item a [[class]] $Obj_{\mathcal{C}}$, called the \textbf{class of [[objects]]};


\item for each $a, b\in Obj_{\mathcal{C}}$, an [[object]]

\begin{displaymath}
\mathcal{C}(a,b)\in \mathcal{V}
  \,,
\end{displaymath}
called the \textbf{$\mathcal{V}$-[[object of morphisms]]} between $a$ and $b$;


\item for each $a,b,c\in Obj(\mathcal{C})$ a [[morphism]] in $\mathcal{V}$

\begin{displaymath}
\circ_{a,b,c}
    \;\colon\;
   \mathcal{C}(a,b)\times \mathcal{C}(b,c)
     \longrightarrow
   \mathcal{C}(a,c)
\end{displaymath}
out of the [[tensor product]] of [[hom-objects]], called the \emph{[[composition]]} operation;


\item for each $a \in Obj(\mathcal{C})$ a morphism $Id_a \colon \ast \to  \mathcal{C}(a,a)$, called the \emph{[[identity]]} morphism on $a$



\end{enumerate}
such that the composition is [[associativity|associative]] and [[unitality|unital]].

If the [[class]] $Obj_{\mathcal{C}}$ happens to be a [[set]] (hence a [[small set]] instead of a [[proper class]]) then we say the $\mathcal{V}$-enriched category $\mathcal{C}$ is \emph{[[small category|small]]}, as in Def. \ref{SmallCategory}.

\end{defn}
\begin{example}
\label{SetEnrichedCategoriesArePlainCategories}\hypertarget{SetEnrichedCategoriesArePlainCategories}{}
\textbf{([[Set]]-[[enriched categories]] are plain [[categories]])}

An [[enriched category]] (Def. \ref{TopEnrichedCategory}) over the [[cosmos]] $\mathcal{V} =$ [[Set]], as in Example \ref{ExamplesOfCosmoi}, is the same as a plain [[category]] (Def. \ref{Categories}).

\end{example}
\begin{example}
\label{CatEnrichedCategoriesAreStrict2Categories}\hypertarget{CatEnrichedCategoriesAreStrict2Categories}{}
\textbf{([[Cat]]-[[enriched categories]] are [[strict 2-categories]])}

An [[enriched category]] (Def. \ref{TopEnrichedCategory}) over the [[cosmos]] $\mathcal{V} =$ [[Cat]], as in Example \ref{ExamplesOfCosmoi}, is the same as a [[strict 2-category]] (Def. \ref{Strict2Categories}).

\end{example}
\begin{example}
\label{UnderlyingCategoryOfTopEnrichedCategory}\hypertarget{UnderlyingCategoryOfTopEnrichedCategory}{}
\textbf{(underlying [[category]] of an [[enriched category]])}

Let $\mathcal{C}$ be a $\mathcal{V}$-[[enriched category]] (Def. \ref{TopEnrichedCategory}).

Using the [[lax monoidal functor|lax monoidal]] [[structure]] (Def. \ref{LaxMonoidalFunctor}) on the [[hom functor]] (Example \ref{ChangeOfCosmosFromVToSet})

\begin{displaymath}
Hom_{\mathcal{V}}(1,-)
  \;\colon\;
  \mathcal{V} \longrightarrow Set
\end{displaymath}
out of the [[tensor unit]] $1 \in \mathcal{C}$ this induces a [[Set]]-[[enriched category]] $\vert \mathcal{C}\vert$ with hence an ordinary [[category]] (Example \ref{SetEnrichedCategoriesArePlainCategories}), with

\begin{itemize}%
\item $Obj_{\vert \mathcal{C}\vert} \;\coloneqq\; Obj_{\mathcal{C}}$;


\item $Hom_{\vert \mathcal{C}\vert}( X, Y )
 \;\coloneqq\;
Hom_{\mathcal{V}}(1, \mathcal{C}(X,Y))$.



\end{itemize}
It is in this sense that $\mathcal{C}$ is a plain [[category]] ${\vert \mathcal{C}\vert}$ equipped with [[extra structure]], and hence an ``[[enriched category]]''.

\end{example}
The archetypical example is $\mathcal{V}$ itself:

\begin{example}
\label{TopkAsATopologicallyEnrichedCategory}\hypertarget{TopkAsATopologicallyEnrichedCategory}{}
\textbf{($\mathcal{V}$ as a $\mathcal{V}$-[[enriched category]])}

Evert [[cosmos]] $\mathcal{C}$ (Def. \ref{Cosmos}) canonically obtains the structure of a $\mathcal{V}$-[[enriched category]], def. \ref{TopEnrichedCategory}:

the [[hom-objects]] are the [[internal homs]]

\begin{displaymath}
\mathcal{v}(X,Y)
    \coloneqq
  [X,Y]
\end{displaymath}
and with [[composition]]

\begin{displaymath}
[X,Y] \times [Y,Z]
    \longrightarrow
  [X,Z]
\end{displaymath}
given by the [[adjunct]] under the ([[Cartesian product]]$\dashv$ [[internal hom]])-[[adjunction]] of the [[evaluation morphisms]]

\begin{displaymath}
X \otimes  [XmY] \otimes [Y,Z]
    \overset{(ev, id)}{\longrightarrow}
  Y \otimes [Y,Z]
    \overset{ev}{\longrightarrow}
  Z
  \,.
\end{displaymath}
\end{example}
The usual construction on categories, such as that of [[opposite categories]] (Def. \ref{OppositeCategory}) and [[product categories]] (Def. \ref{ProductCategory}) have evident enriched analogs

\begin{defn}
\label{OppositeAndProductOfPointedTopologicallyEnrichedCategory}\hypertarget{OppositeAndProductOfPointedTopologicallyEnrichedCategory}{}
\textbf{([[enriched category|enriched]] [[opposite category]] and [[product category]])}

For $\mathcal{V}$ a [[cosmos]], let $\mathcal{C}, \mathcal{D}$ be $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}).

\begin{enumerate}%
\item The \textbf{[[opposite category|opposite]] [[enriched category]]} $\mathcal{C}^{op}$ is the [[enriched category]] with the same [[objects]] as $\mathcal{C}$, with [[hom-objects]]

\begin{displaymath}
\mathcal{C}^{op}(X,Y)
  \coloneqq
  \mathcal{C}(Y,X)
\end{displaymath}
and with [[composition]] given by [[braiding]] \eqref{BraidingIsomorphism} followed by the [[composition]] in $\mathcal{C}$:

\begin{displaymath}
\mathcal{C}^{op}(X,Y)
  \otimes
  \mathcal{C}^{op}(Y,Z)
  =
  \mathcal{C}(Y,X) \otimes \mathcal{C}(Z,Y)
  \underoverset{\simeq}{\tau}{\longrightarrow}
  \mathcal{C}(Z,Y) \otimes \mathcal{C}(Y,X)
    \overset{\circ_{Z,Y,X}}{\longrightarrow}
  \mathcal{C}(Z,X)
  =
  \mathcal{C}^{op}(X,Z)
  \,.
\end{displaymath}

\item the \textbf{[[enriched category|enriched]] [[product category]]} $\mathcal{C} \times \mathcal{D}$ is the [[enriched category]] whose [[objects]] are [[pairs]] of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, whose [[hom-spaces]] are the [[tensor product]] of the separate [[hom objects]]

\begin{displaymath}
(\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) )
    \coloneqq
  \mathcal{C}(c_1,c_2)\otimes \mathcal{D}(d_1,d_2)
\end{displaymath}
and whose [[composition]] operation is the [[braiding]] \eqref{BraidingIsomorphism} followed by the [[tensor product]] of the separate composition operations:

\begin{displaymath}
\itexarray{
    (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2))
      \otimes
    (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3))
    \\
    {}^{\mathllap{=}}\downarrow
    \\
    \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{D}(d_1,d_2)\right)
      \otimes
    \left(\mathcal{C}(c_2,c_3) \otimes \mathcal{D}(d_2,d_3)\right)
    \\
    \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}}
    \\
    \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{C}(c_2,c_3)\right)
      \otimes
    \left( \mathcal{D}(d_1,d_2) \otimes  \mathcal{D}(d_2,d_3)\right)
      &\overset{
    (\circ_{c_1,c_2,c_3}) \otimes (\circ_{d_1,d_2,d_3})
    }{\longrightarrow}
      &
    \mathcal{C}(c_1,c_3) \otimes \mathcal{D}(d_1,d_3)
    \\
    && \downarrow^{\mathrlap{=}}
    \\
    && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3))
  }
 \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{defn}
\label{TopologicallyEnrichedFunctor}\hypertarget{TopologicallyEnrichedFunctor}{}
\textbf{([[enriched functor]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$ and $\mathcal{D}$ be two $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}).

A \emph{$\mathcal{V}$-[[enriched functor]]} from $\mathcal{C}$ to $\mathcal{D}$

\begin{displaymath}
F \;\colon\;  \mathcal{C}  \longrightarrow \mathcal{D}
\end{displaymath}
is

\begin{enumerate}%
\item a [[function]]

\begin{displaymath}
F_{Obj} \;\colon\; Obj_{\mathcal{C}} \longrightarrow Obj_{\mathcal{D}}
\end{displaymath}
of [[objects]];


\item for each $a,b \in Obj_{\mathcal{C}}$ a [[morphism]] in $\mathcal{V}$

\begin{displaymath}
F_{a,b}
    \;\colon\;
  \mathcal{C}(a,b)
    \longrightarrow
  \mathcal{D}(F_0(a), F_0(b))
\end{displaymath}
between [[hom-objects]]



\end{enumerate}
such that this preserves [[composition]] and [[identity]] morphisms in the evident sense.

\end{defn}
\begin{example}
\label{EnrichedHomFunctor}\hypertarget{EnrichedHomFunctor}{}
\textbf{([[enriched hom-functor]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$ be a $\mathcal{V}$-[[enriched category]] (Def. \ref{TopEnrichedCategory}). Then there is a $\mathcal{V}$-[[enriched functor]] out of the enriched [[product category]] of $\mathcal{C}$ with its enriched [[opposite category]] (Def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory})

\begin{displaymath}
\mathcal{C}(-,-)
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  \mathcal{V}
\end{displaymath}
to $\mathcal{V}$, regarded as a $\mathcal{V}$-[[enriched category]] (Example \ref{TopkAsATopologicallyEnrichedCategory}), which sends a [[pair]] of [[objects]] $X,Y \in \mathcal{C}$ to the [[hom-object]] $\mathcal{C}(X,Y) \in \mathcal{V}$, and which acts on morphisms by [[composition]] in the evident way.

\end{example}
\begin{example}
\label{EnrichedPresheaf}\hypertarget{EnrichedPresheaf}{}
\textbf{([[enriched presheaves]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$ be a $\mathcal{V}$-[[enriched category]] (Def. \ref{TopEnrichedCategory}). Then a $\mathcal{V}$-[[enriched functor]] (Def. \ref{TopologicallyEnrichedFunctor})

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{V}
\end{displaymath}
to the archetypical $\mathcal{V}$-[[enriched category]] from Example \ref{TopkAsATopologicallyEnrichedCategory} is:

\begin{enumerate}%
\item an [[object]] $F_a \in Obj_{\mathcal{V}}$ for each object $a \in Obj_{\mathcal{C}}$;


\item a [[morphism]] in $\mathcal{V}$ of the form

\begin{displaymath}
F_a \otimes \mathcal{C}(a,b) \longrightarrow F_b
\end{displaymath}
for all pairs of objects $a,b \in Obj(\mathcal{C})$

(this is the [[adjunct]] of $F_{a,b}$ under the [[adjunction]] \eqref{InternalHomAdjunction} on $\mathcal{V}$)



\end{enumerate}
such that composition is respected, in the evident sense.

For every object $c \in \mathcal{C}$, there is an enriched [[representable functor]], denoted

\begin{displaymath}
y(c) \;\coloneqq\; \mathcal{C}(c,-)
\end{displaymath}
(where on the right we have the [[enriched hom-functor]] from Example \ref{EnrichedHomFunctor})

which sends objects to

\begin{displaymath}
y(c)(d) = \mathcal{C}(c,d) \in \mathcal{V}
\end{displaymath}
and whose action on morphisms is, under the above identification, just the [[composition]] operation in $\mathcal{C}$.

\end{example}
More generally, the following situation will be of interest:

\begin{example}
\label{PointedTopologicalFunctorOnProductCategory}\hypertarget{PointedTopologicalFunctorOnProductCategory}{}
\textbf{([[enriched functor]] on [[enriched category|enriched]] [[product category]] with [[opposite category]])}

An $\mathcal{V}$-[[enriched functor]] (Def. \ref{TopologicallyEnrichedFunctor}) into $\mathcal{V}$ (Example \ref{TopkAsATopologicallyEnrichedCategory}) out of an [[enriched category|enriched]] [[product category]] (Def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory})

\begin{displaymath}
F
    \;\colon\;
  \mathcal{C} \times \mathcal{D}
    \longrightarrow
  \mathcal{V}
\end{displaymath}
(an ``enriched [[bifunctor]]'') has component morphisms of the form

\begin{displaymath}
F_{(c_1,d_1),(c_2,d_2)}
    \;\colon\;
  \mathcal{C}(c_1,c_2)
  \otimes
  \mathcal{D}(d_1,d_2)
    \longrightarrow
  \big[ F_0((c_1,d_1)),F_0((c_2,d_2)) \big]
  \,.
\end{displaymath}
By functoriality and under passing to [[adjuncts]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) under \eqref{InternalHomAdjunction} this is equivalent to two commuting \emph{[[actions]]}

\begin{displaymath}
\rho_{c_1,c_2}(d)
   \;\colon\;
  \mathcal{C}(c_1,c_2) \otimes F_0((c_1,d))
    \longrightarrow
  F_0((c_2,d))
\end{displaymath}
and

\begin{displaymath}
\rho_{d_1,d_2}(c)
    \;\colon\;
  \mathcal{D}(d_1,d_2) \otimes F_0((c,d_1))
    \longrightarrow
  F_0((c,d_2))
  \,.
\end{displaymath}
In the special case of a functor out of the [[enriched category|enriched]] [[product category]] of some $\mathcal{V}$-[[enriched category]] $\mathcal{C}$ with its enriched [[opposite category]] (def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory})

\begin{displaymath}
F
    \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  \mathcal{V}
\end{displaymath}
then this takes the form of a ``pullback action'' in the first variable

\begin{displaymath}
\rho_{c_2,c_1}(d)
   \;\colon\;
  \mathcal{C}(c_1,c_2) \otimes F_0((c_2,d))
    \longrightarrow
  F_0((c_1,d))
\end{displaymath}
and a ``pushforward action'' in the second variable

\begin{displaymath}
\rho_{d_1,d_2}(c)
    \;\colon\;
  \mathcal{C}(d_1,d_2) \otimes F_0((c,d_1))
    \longrightarrow
  F_0((c,d_2))
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{EnrichedNaturalTransformation}\hypertarget{EnrichedNaturalTransformation}{}
\textbf{([[enriched natural transformation]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$ and $\mathcal{D}$ be two $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}) and let

\begin{displaymath}
\mathcal{C}
    \underoverset
      {\underset{G}{\longrightarrow}}
      {\overset{F}{\longrightarrow}}
      {}
  \mathcal{D}
\end{displaymath}
be two $\mathcal{V}$-[[enriched functors]] (Def. \ref{TopologicallyEnrichedFunctor}) from $\mathcal{C}$ to $\mathcal{D}$.

Then a $\mathcal{V}$-[[enriched natural transformation]]

\begin{displaymath}
\mathcal{C}
    \underoverset
      {\underset{G}{\longrightarrow}}
      {\overset{F}{\longrightarrow}}
      {\phantom{AA}\Downarrow{\mathrlap{\eta}}\phantom{AA}}
  \mathcal{D}
\end{displaymath}
is

\begin{itemize}%
\item for each $c \in Obj_{\mathcal{C}}$ a choice of [[morphism]]

\begin{displaymath}
\eta_c \;\colon\; I \longrightarrow \mathcal{D}(F(c),G(c))
\end{displaymath}
such that for each [[pair]] of [[objects]] $c,d \in \mathcal{C}$ the two [[morphisms]] (in $\mathcal{V}$)



\end{itemize}
\begin{equation}
\eta_d \circ F(-)
   \;\colon\;
  \mathcal{C}(c,d)
    \overset{r}{\simeq}
  \mathcal{C}(c,d) \otimes I
    \overset{ G_{c,d} \otimes \eta_c }{\longrightarrow}
  \mathcal{D}(G(c),G(d)) \otimes \mathcal{D}( F(c), G(c) )
    \overset{\circ_{F(c), G(c), G(d)}}{\longrightarrow}
  \mathcal{D}(F(c), G(d))
\label{OneComposite}\end{equation}
and

\begin{equation}
G(-) \circ \eta_c
    \;\colon\;
  \mathcal{C}(c,d)
    \overset{\ell}{\simeq}
  I \otimes  \mathcal{C}(c,d)
    \overset{ \eta_d \otimes F_{c,d} }{\longrightarrow}
  \mathcal{D}(F(d), G(d)) \otimes \mathcal{D}(F(c), F(d))
    \overset{\circ_{F(c),F(d), G(d)}}{\longrightarrow}
  \mathcal{D}(F(c), G(d))
\label{OtherComposite}\end{equation}
agree.

\end{defn}
\begin{example}
\label{CategoryOfEnrichedFunctors}\hypertarget{CategoryOfEnrichedFunctors}{}
\textbf{([[functor category]] of [[enriched functors]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}) let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}). Then there is a [[category]] (Def. \ref{Categories}) of [[enriched functors]] (Def. \ref{TopologicallyEnrichedFunctor}), to be denoted

\begin{displaymath}
[\mathcal{C}, \mathcal{D}]
\end{displaymath}
whose [[objects]] are the [[enriched functors]] $\mathcal{C} \overset{F}{\to} \mathcal{D}$ and whose [[morphisms]] are the [[enriched natural transformations]] between these (Def. \ref{EnrichedNaturalTransformation}).

In the case that $\mathcal{V} =$ [[Set]], via Def. \ref{ExamplesOfCosmoi}, with $Set$-enriched categories identified with plain categories via Example \ref{SetEnrichedCategoriesArePlainCategories}, this coincides with the [[functor category]] from Example \ref{FunctorCategory}.

Notice that, at this point, $[\mathcal{C}, \mathcal{D}]$ is a plain [[category]], not itself a $\mathcal{V}$-[[enriched category]], unless $\mathcal{V} =$ [[Set]]. But it may be enhanced to one, this is Def. \ref{PointedTopologicalFunctorCategory} below.

\end{example}
There is now the following evident generalization of the concept of \emph{[[adjoint functors]]} (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) from plain [[category theory]] to [[enriched category theory]]:

\begin{defn}
\label{EnrichedAdjunction}\hypertarget{EnrichedAdjunction}{}
\textbf{([[enriched adjunction]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}). Then an \emph{[[adjoint pair]] of $\mathcal{V}$-[[enriched functors]]} or \emph{[[enriched adjunction]]}

\begin{displaymath}
\mathcal{C}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot}
  \mathcal{D}
\end{displaymath}
is a [[pair]] of $\mathcal{V}$-[[enriched functors]] (Def. \ref{TopologicallyEnrichedFunctor}), as shown, such that there is a $\mathcal{V}$-[[enriched natural isomorphism]] (Def. \ref{EnrichedNaturalTransformation}) between [[enriched hom-functors]] (Def. \ref{EnrichedHomFunctor}) of the form

\begin{equation}
\mathcal{C}(L(-),-)
  \;\simeq\;
  \mathcal{D}(-,R(-))
  \,.
\label{EnrichedAdjunctionIsomorphism}\end{equation}
\end{defn}
\begin{defn}
\label{EnrichedEquivalenceOfCategories}\hypertarget{EnrichedEquivalenceOfCategories}{}
\textbf{([[enriched category theory|enriched]] [[equivalence of categories]])}

For $\mathcal{V}$ a [[cosmos]] (Def. \ref{Cosmos}), let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-[[enriched categories]] (Def. \ref{TopEnrichedCategory}). Then an \emph{equivalence of enriched categories}

\begin{displaymath}
\itexarray{
    \mathcal{C}
      \underoverset
        {\underset{R}{\longrightarrow}}
        {\overset{L}{\longleftarrow}}
        {\phantom{AA} \simeq \phantom{AA}}
    \mathcal{D}
  }
\end{displaymath}
is a [[pair]] of $\mathcal{V}$-[[enriched functors]] back and forth, as shown (Def. \ref{TopologicallyEnrichedFunctor}), together with $\mathcal{V}$-[[enriched natural isomorphisms]] (Def. \ref{EnrichedNaturalTransformation}) between their [[composition]] and the [[identity functors]]:

\begin{displaymath}
id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L
  \phantom{AAA}
   \text{and}
  \phantom{AAA}
  L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}}
  \,.
\end{displaymath}
\end{defn}


\end{document}