nLab geometry of physics - universal constructions

Universal constructions

What makes category theory be theory, as opposed to just a language, is the concept of universal constructions. This refers to the idea of objects with a prescribed property which are universal with this property, in that they “know about” or “subsume” every other object with that same kind of property. Category theory allows to make precise what this means, and then to discover and prove theorems about it.

Universal constructions are all over the place in mathematics. Iteratively finding the universal constructions in a prescribed situation essentially amounts to systematically following the unravelling of the given situation or problem or theory that one is studying.

There are several different formulations of the concept of universal constructions, discussed below:

• Limits and colimits

• Ends and coends

• Left and right Kan extensions

But these three kinds of constructions all turn out to be special cases of each other, hence they really reflect different perspectives on a single topic of universal constructions. In fact, all three are also special cases of the concept of adjunction (Def. ), thus re-amplifying that category theory is really the theory of adjunctions and hence, if we follow (Lambek 82), of duality.

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Limits and colimits

Maybe the most hands-on version of universal constructions are limits (Def. below), which is short for limiting cones (Remark below). The formally dual concept (Example ) is called colimits (which are hence limits in an opposite category). Other terminology is in use, too:

$\phantom{A}$ $\underset{\longleftarrow}{\lim}$ $\phantom{A}$	$\phantom{A}$ $\underset{\longrightarrow}{\lim}$ $\phantom{A}$
$\phantom{A}$ limit $\phantom{A}$	$\phantom{A}$ colimit $\phantom{A}$
$\phantom{A}$ inverse limit $\phantom{A}$	$\phantom{A}$ direct limit$\phantom{A}$

There is a variety of different kinds of limits/colimits, depending on the diagram shape that they are limiting (co-)cones over. This includes universal constructions known as equalizers, products, fiber products/pullbacks, filtered limits and various others, all of which are basic tools frequently used whenever category theory applies.

A key fact of category theory, regarding limits, is that right adjoints preserve limits and left adjoints preserve colimits (Prop. below). This will be used all the time. A partial converse to this statement is that if a functor preserves limits/colimits, then its adjoint functor is, if it exists, objectwise given by a limit/colimit over a comma category under/over the given functor (Prop. below). Since these comma categories are in general not small, this involves set-theoretic size subtleties that are dealt with by the adjoint functor theorem (Remark below). We discuss in detail a very special but also very useful special case of this in Prop. , further below.

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Definition

(limit and colimit)

Let $\mathcal{C}$ be a small category (Def. ), and let $\mathcal{D}$ be any category (Def. ). In this case one also says that a functor

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

is a diagram of shape $\mathcal{C}$ in $\mathcal{D}$.

Recalling the functor category (Example ) $[\mathcal{C}, \mathcal{D}]$, there is the constant diagram-functor

$$const \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}, \mathcal{D}]$$

which sends an object $X \in \mathcal{D}$ to the functor that sends every $c \in \mathcal{C}$ to $X$, and every morphism in $\mathcal{C}$ to the identity morphism on $X$. Accordingly, every morphism in $\mathcal{D}$ is sent by $const$ to the natural transformation (Def. ) all whose components are equal to that morphism.

Now:

1. if $const$ has a right adjoint (Def. ), this is called the construction of forming the limiting cone of $\mathcal{C}$-shaped diagrams in $\mathcal{D}$, or just limit (or inverse limit) for short, and denoted

   $$\underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \;\colon\; [\mathcal{C}, \mathcal{D}] \longrightarrow \mathcal{D}$$

2. if $const$ has a left adjoint (Def. ), this is called the construction of forming the colimiting cocone of $\mathcal{C}$-shaped diagrams in $\mathcal{D}$, or just colimit (or direct limit) for short, and denoted

   $$\underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim} \;\colon\; [\mathcal{C}, \mathcal{D}] \longrightarrow \mathcal{D}$$

(1)$$[\mathcal{C}, \mathcal{D}] \array{ \overset{ \phantom{AA}\underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim} \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} const \phantom{AA} }{ \longleftarrow } \\ \overset{ \phantom{AA} \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \phantom{AA}}{\longrightarrow} } \mathcal{D} \,.$$

If $\underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim}$ ($\underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim}$) exists for a given $\mathcal{D}$, one says that $\mathcal{D}$ has all limits (_has all colimits_) of shape $\mathcal{C}$_ or that all limits (colimits) of shape $\mathcal{D}$ exist in $\mathcal{D}$. If this is the case for all small diagrams $\mathcal{C}$, one says that $\mathcal{D}$ has all limits (_has all colimits_) or that all limits exist in $\mathcal{D}$, (_all colimits exist in $\mathcal{D}$.)

Remark

(limit cones)

Unwinding Definition of limits and colimits, it says the following.

First of all, for $d \in \mathcal{D}$ any object and $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ any functor, a natural transformation (Def. ) of the form

(2)$$const_d \overset{i}{\Rightarrow} F$$

has component morphisms

$$\array{ d \\ \big\downarrow^{\mathrlap{i_c}} \\ F(c) }$$

in $\mathcal{D}$, for each $c \in \mathcal{C}$, and the naturality condition (?) says that these form a commuting diagram (Def. ) of the form

(3)$$\array{ && d \\ & {}^{\mathllap{ i_{c_1} } }\swarrow && \searrow^{\mathrlap{ i_{c_2} }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} && F(c_2) }$$

for each morphism $c_1 \overset{f}{\to} c_2$ in $\mathcal{C}$. Due to the look of this diagram, one also calls such a natural transformation a cone over the functor $F$.

Now the counit (Def. ) of the $(const \dashv \underset{\longleftarrow}{\lim})$-adjunction (1) is a natural transformation of the form

$$const_{\underset{\longleftarrow}{\lim} F} \overset{ \phantom{AA} \epsilon_{F} \phantom{AA} }{\longrightarrow} F$$

and hence is, in components, a cone (3) over $F$:

(4)$$\array{ && \underset{\longleftarrow}{\lim} F \\ & {}^{\mathllap{ \epsilon_F(c_1) } }\swarrow && \searrow^{\mathrlap{ \epsilon_F(c_2) }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} && F(c_2) }$$

to be called the limiting cone over $F$

But the universal property of adjunctions says that this is a very special cone: By Prop. the defining property of the limit is equivalently that for every natural transformation of the form (2), hence for every cone of the form (3), there is a unique natural transformation

$$\array{ const_d &\overset{\widetilde i}{\Rightarrow}& const_{ \underset{ \longleftarrow }{\lim} } }$$

which, due to constancy of the two functors applied in the naturality condition (?), has a constant component morphism

(5)$$d \overset{ \widetilde i }{\longrightarrow} \underset{\longleftarrow}{\lim} F$$

such that

$$\array{ const_d && \overset{\widetilde i}{\longrightarrow} && const_{ \underset{\longleftarrow}{\lim} F } \\ & {}_{\mathllap{ \epsilon_F }} \searrow && \swarrow_{ \mathrlap{i} } \\ && F }$$

hence such that (5) factors the given cone (3) through the special cone (4):

$$\array{ && d \\ & {}^{\mathllap{ i_{c_1} } }\swarrow && \searrow^{\mathrlap{ i_{c_2} }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} F(c_2) } \phantom{AAA} = \phantom{AAA} \array{ && d \\ && \big\downarrow^{ \mathrlap{ \widetilde i } } \\ && \underset{\longleftarrow}{\lim} F \\ & {}^{\mathllap{ \epsilon_F(c_1) } }\swarrow && \searrow^{\mathrlap{ \epsilon_F(c_2) }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} F(c_2) }$$

In this case one also says that $\widetilde i$ is a morphism of cones.

Hence a limit cone is a cone over $F$, such that every other cone factors through it in a unique way.

Of course this concept of (co)limiting cone over a functor $F \;\colon\; \mathcal{C} \to \mathcal{D}$ makes sense also when

1. $\mathcal{C}$ is not small,

2. and/or when a (co-)limiting cone exists only for some but not for all functors of this form.

Example

(terminal/initial object is empty limit/colimit)

Let $\mathcal{C}$ be a category, and let $\ast \in \mathcal{C}$ be an object. The following are equivalent:

1. $\ast$ is a terminal object of $\mathcal{C}$ (Def. );

2. $\ast$ is the limit of the empty diagram.

And formally dual (example ): Let $\emptyset \in \mathcal{C}$ be an object. The following are equivalent:

1. $\emptyset$ is an initial object of $\mathcal{C}$ (Def. );

2. $\emptyset$ is the colimit of the empty diagram.

Proof

We discuss the case of the terminal object, the other case is formally dual (Example ).

It suffices to observe that a cone over the empty diagram (Remark ) is clearly just a plain object of $\mathcal{C}$. Hence a morphism of such cones is just a plain morphism of $\mathcal{C}$. This way the condition on a limiting cone is now manifestly the same as the condition on a terminal object.

Example

(initial object is limit over identity functor)

Let $\mathcal{C}$ be a category, and let $\emptyset \in \mathcal{C}$ be an object. The following are equivalent:

1. $\emptyset$ is an initial object of $\mathcal{C}$ (Def. );

2. $\emptyset$ is the tip of a limit cone (Remark ) over the identity functor on $\mathcal{C}$.

Proof

First let $\emptyset$ be an initial object. Then, by definition, it is the tip of a unique cone over the identity functor

(6)$$\array{ const_{\emptyset}&\phantom{AA}& && \emptyset \\ {}^{\mathllap{i^{\emptyset}}}\Downarrow && & {}^{\mathllap{i^{\emptyset}_{c_1}}}\swarrow && \searrow^{\mathrlap{i^{\emptyset}_{c_2}}} \\ id_{\mathcal{C}} && c_1 && \underset{f}{\longrightarrow} && c_2 }$$

We need to show that that every other cone $i^x$

$$\array{ const_{x}&\phantom{AA}& && x \\ {}^{\mathllap{\mathllap{i^x}}}\Downarrow && & {}^{i^x_{c_1}}\swarrow && \searrow^{\mathrlap{i^x_{c_2}}} \\ id_{\mathcal{C}} && c_1 && \underset{f}{\longrightarrow} && c_2 }$$

factors uniquely through $i^\emptyset$.

First of all, since the cones are over the identity functor, there is the component $i^x_{\emptyset} \;\colon\; x \to \emptyset$, and it is a morphism of cones.

To see that this is the unique morphism of cones, consider any morphism of cones $j^x_\emptyset$, hence a morphism in $\mathcal{C}$ such that $i^x_c = i^\emptyset_c \circ j^x_\emptyset$ for all $c \in \mathcal{C}$. Taking here $c = \emptyset$ yields

$$\begin{aligned} i^x_\emptyset & = \underset{ = id_\emptyset }{\underbrace{i^\emptyset_{\emptyset}}} \circ j^x_{\emptyset} \\ & = j^x_\emptyset \,, \end{aligned}$$

where under the brace we used that $\emptyset$ is initial. This proves that $i^\emptyset$ is the limiting cone.

For the converse, assume now that $i^\emptyset$ is a limiting cone over the identity functor, with labels as in (6). We need to show that its tip $\emptyset$ is an initial object.

Now the cone condition applied for any object $x \in \mathcal{C}$ over the morphims $f \coloneqq i^\emptyset_x$ says that

$$i^{\emptyset}_x \circ i^\emptyset_\emptyset = i^\emptyset_x$$

which means that $i^\emptyset_\emptyset$ constitutes a morphism of cones from $i^\emptyset$ to itself. But since $i^\emptyset$ is assumed to be a limiting cone, and since the identity morphism on $\emptyset$ is of course also a morphism of cones from $i^\emptyset$ to itsely, we deduce that

(7)$$i^\emptyset_\emptyset \;=\; id_{\emptyset} \,.$$

Now consider any morphism of the form $\emptyset \overset{f}{\to} x$. Since we already have the morphism $\emptyset \overset{i^\emptyset_x}{\to} x$, to show initiality of $\emptyset$ we need to show that $f = i^\emptyset_x$.

Indeed, the cone condition of $i^\emptyset_x$ applied to $f$ now yields

$$\begin{aligned} i^\emptyset_x & = f \circ \underset{ = id_{\emptyset} }{\underbrace{i^\emptyset_\emptyset}} \\ & = f\,, \end{aligned}$$

where under the brace we used (7).

Example

(limits of presheaves are computed objectwise)

Let $\mathcal{C}$ be a category and write $[\mathcal{C}^{op}, Set]$ for its category of presheaves (Example ). Let moreover $\mathcal{D}$ be a small category and consider any functor

$$F \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}] \,,$$

hence a $\mathcal{D}$-shaped diagram in the category of presheaves.

Then

1. The limit (Def. ) of $F$ exists, and is the presheaf which over any object $c \in \mathcal{C}$ is given by the limit in Set of the values of the presheaves at $c$:

   $$\left( \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d)(c)$$

2. The colimit (Def. ) of $F$ exists, and is the presheaf which over any object $c \in \mathcal{C}$ is given by the colimit in Set of the values of the presheaves at $c$:

   $$\left( \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d)(c)$$

Proof

We discuss the case of limits, the other case is formally dual (Example ).

Observe that there is a canonical equivalence (Def. )

$$[\mathcal{D}, [\mathcal{C}^{op}, \Set]] \simeq [\mathcal{D} \times \mathcal{C}^{op}, Set]$$

where $\mathcal{D} \times \mathcal{C}^{op}$ is the product category.

This makes manifest that a functor $F \;\colon\; \mathcal{D} \to [\mathcal{C}^{op}, Set]$ is equivalently a diagram of the form

$$\array{ && \vdots && && \vdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ && \vdots && && \vdots }$$

Then observe that taking the limit of each “horizontal row” in such a diagram indead does yield a presheaf on $\mathcal{C}$, in that the construction extends from objects to morphisms, and uniquely so: This is because for any morphism $c_1 \overset{g}{\to} c_2$ in $\mathcal{C}$, a cone over $F(-)(c_2)$ (Remark ) induces a cone over $F(-)(c_1)$, by vertical composition with $F(-)(g)$

$$\array{ && \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ & {}^{ }\swarrow && \searrow \\ F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) \\ {}^{\mathllap{F(d_1)(g)}}\big\downarrow && && \big\downarrow^{\mathrlap{F(d_2)(g)}} \\ F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) }$$

From this, the universal property of limits of sets (as in Remark ) implies that there is a unique morphism between the pointwise limits which constitutes a presheaf over $\mathcal{C}$

$$\array{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow^{\mathrlap{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(g) }} \\ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) }$$

and that is the tip of a cone over the diagram $F(-)$ in presheaves.

Hence it remains to see that this cone of presheaves is indeed universal.

Now if $I$ is any other cone over $F$ in the category of presheaves, then by the universal property of the pointswise limits, there is for each $c \in \mathcal{C}$ a unique morphism of cones in sets

$$I(c) \longrightarrow \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c) \,.$$

Hence there is at most one morphisms of cones of presheaves, namely if these components make all their naturality squares commute.

$$\array{ I(c_2) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow && \big\downarrow \\ I(c_1) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) } \,.$$

But since everything else commutes, the two ways of going around this diagram constitute two morphisms from a cone over $F(-)(c_1)$ to the limit cone over $F(-)(c_1)$, and hence they must be equal, by the universal property of limits.

Proposition

(hom-functor preserves limits)

Let $\mathcal{C}$ be a category and write

$$Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set$$

for its hom-functor. This preserves limits (Def. ) in both its arguments (recalling that a limit in the opposite category $\mathcal{C}^{op}$ is a colimit in $\mathcal{C}$).

More in detail, let $X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C}$ be a diagram. Then:

1. If the limit $\underset{\longleftarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a natural isomorphism

   $$Hom_{\mathcal{C}}\left(Y, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( Y, X_i \right) \right) \,,$$

   where on the right we have the limit over the diagram of hom-sets given by

   $$Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-) }{\longrightarrow} Set\,.$$

2. If the colimit $\underset{\longrightarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a natural isomorphism

   $$Hom_{\mathcal{C}}\left(\underset{\longrightarrow}{\lim}_i X_i ,Y\right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( X_i , Y\right) \right) \,,$$

   where on the right we have the limit over the diagram of hom-sets given by

   $$Hom_{\mathcal{C}}(-,Y) \circ X \;\colon\; \mathcal{I}^{op} \overset{X}{\longrightarrow} \mathcal{C}^{op} \overset{Hom_{\mathcal{C}}(-,Y) }{\longrightarrow} Set\,.$$

Proof

We give the proof of the first statement, the proof of the second statement is formally dual (Example ).

First observe that, by the very definition of limiting cones, maps out of some $Y$ into them are in natural bijection with the set $Cones\left(Y, X_\bullet \right)$ of cones over the diagram $X_\bullet$ with tip $Y$:

$$Hom\left( Y, \underset{\longleftarrow}{\lim}_{i} X_i \right) \;\simeq\; Cones\left( Y, X_\bullet \right) \,.$$

Hence it remains to show that there is also a natural bijection like so:

$$Cones\left( Y, X_\bullet \right) \;\simeq\; \underset{\longleftarrow}{\lim}_{i} \left( Hom(Y,X_i) \right) \,.$$

Now, again by the very definition of limiting cones, a single element in the limit on the right is equivalently a cone of the form

$$\left\{ \array{ && \ast \\ & {}^{\mathllap{const_{p_i}}}\swarrow && \searrow^{\mathrlap{const_{p_j}}} \\ Hom(Y,X_i) && \underset{X_\alpha \circ (-)}{\longrightarrow} && Hom(Y,X_j) } \right\}_{i, j \in Obj(\mathcal{I}), \alpha \in Hom_{\mathcal{I}}(i,j) } \,.$$

This is equivalently for each object $i \in \mathcal{I}$ a choice of morphism $p_i \colon Y \to X_i$ , such that for each pair of objects $i,j \in \mathcal{I}$ and each $\alpha \in Hom_{\mathcal{I}}(i,j)$ we have $X_\alpha \circ p_i = p_j$. And indeed, this is precisely the characterization of an element in the set $Cones\left( Y, X_\bullet\} \right)$.

Example

(initial and terminal object in terms of adjunction)

Let $\mathcal{C}$ be a category (Def. ).

1. The following are equivalent:

   1. $\mathcal{C}$ has a terminal object (Def. );

   2. the unique functor $\mathcal{C} \to \ast$ (Def. ) to the terminal category (Example ) has a right adjoint (Def. )

      $$\ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}$$

   Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.

2. Dually, the following are equivalent:

   1. $\mathcal{C}$ has an initial object (Def. );

   2. the unique functor $\mathcal{C} \to \ast$ to the terminal category has a left adjoint

      $$\mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast$$

   Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.

Proof

Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism (?) characterizing the adjoint functors is directly the universal property of an initial object in $\mathcal{C}$

$$Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast$$

or of a terminal object

$$Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,,$$

respectively.

Proposition

(left adjoints preserve colimits and right adjoints preserve limits)

Let $(L \dashv R) \colon \mathcal{D} \to \mathcal{C}$ be a pair of adjoint functors (Def. ). Then

• $L$ preserves all colimits (Def. ) that exist in $\mathcal{C}$,

• $R$ preserves all limits (Def. ) in $\mathcal{D}$.

Proof

Let $y : I \to \mathcal{D}$ be a diagram whose limit $\lim_{\leftarrow_i} y_i$ exists. Then we have a sequence of natural isomorphisms, natural in $x \in C$

$$\begin{aligned} Hom_{\mathcal{C}}(x, R {\lim_\leftarrow}_i y_i) & \simeq Hom_{\mathcal{D}}(L x, {\lim_\leftarrow}_i y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{D}}(L x, y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{C}}( x, R y_i) \\ & \simeq Hom_{\mathcal{C}}( x, {\lim_\leftarrow}_i R y_i) \,, \end{aligned}$$

where we used the hom-isomorphism (?) and the fact that any hom-functor preserves limits (Def. ). Because this is natural in $x$ the Yoneda lemma implies that we have an isomorphism

$$R {\lim_\leftarrow}_i y_i \simeq {\lim_\leftarrow}_i R y_i \,.$$

The argument that shows the preservation of colimits by $L$ is analogous.

Proposition

(limits commute with limits)

Let $\mathcal{D}$ and $\mathcal{D}'$ be small categories (Def. ) and let $\mathcal{C}$ be a category (Def. ) which admits limits (Def. ) of shape $\mathcal{D}$ as well as limits of shape $\mathcal{D}'$. Then these limits “commute” with each other, in that for $F \;\colon\; \mathcal{D} \times {\mathcal{D}'} \to \mathcal{C}$ a functor (hence a diagram of shape the product category), with corresponding adjunct functors (via Example )

$${\mathcal{D}'} \overset{F_{\mathcal{D}}}{\longrightarrow} [\mathcal{D},\mathcal{C}] \phantom{AAA} {\mathcal{D}} \overset{F_{\mathcal{D}'}}{\longrightarrow} [{\mathcal{D}'}, \mathcal{C}]$$

we have that the canonical comparison morphism

(8)$$lim F \simeq lim_{\mathcal{D}} (lim_{\mathcal{D}'} F_{\mathcal{D}} ) \simeq lim_{\mathcal{D}'} (lim_{\mathcal{D}} F_{\mathcal{D}'} )$$

is an isomorphism.

Proof

Since the limit-construction is the right adjoint functor to the constant diagram-functor, this is a special case of right adjoints preserve limits (Prop. ).

See limits and colimits by example for what formula (8) says for instance for the special case $\mathcal{C} =$ Set.

Remark

(general non-commutativity of limits with colimits)

In general limits do not commute with colimits. But under a number of special conditions of interest they do. Special cases and concrete examples are discussed at commutativity of limits and colimits.

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Proposition

(pointwise expression of left adjoints in terms of limits over comma categories)

A functor $R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ (Def. ) has a left adjoint $L \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ (Def. ) precisely if

1. $R$ preserves all limits (Def. ) that exist in $\mathcal{C}$;

2. for each object $d \in \mathcal{D}$, the limit (Def. ) of the canonical functor (?) out of the comma category (Example )

   $$d/R \longrightarrow \mathcal{C}$$

   exists.

In this case the value of the left adjoint $L$ on $d$ is given by that limit:

(9)$$L(d) \;\simeq\; \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c$$

Proof

First assume that the left adjoint exist. Then

1. $R$ is a right adjoint and hence preserves limits since all right adjoints preserve limits (Prop. );

2. by Prop. the adjunction unit provides a universal morphism $\eta_d$ into $L(d)$, and hence, by Prop. , exhibits $(L(d), \eta_d)$ as the initial object of the comma category $d/R$. The limit over any category with an initial object exists, as it is given by that initial object.

Conversely, assume that the two conditions are satisfied and let $L(d)$ be given by (9). We need to show that this yields a left adjoint.

By the assumption that $R$ preserves all limits that exist, we have

(10)$$\array{ R(L(d)) & = R\left( \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c \right) \\ & \simeq \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} R(c) }$$

Since the $d \overset{f}{\to} R(d)$ constitute a cone over the diagram of the $R(d)$, there is universal morphism

$$d \overset{\phantom{AA} \eta_d \phantom{AA}}{\longrightarrow} R(L(d)) \,.$$

By Prop. it is now sufficient to show that $\eta_d$ is a universal morphism into $L(d)$, hence that for all $c \in \mathcal{C}$ and $d \overset{g}{\longrightarrow} R(c)$ there is a unique morphism $L(d) \overset{\widetilde f}{\longrightarrow} c$ such that

$$\array{ && d \\ & {}^{\mathllap{ \eta_d }}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(d)) && \underset{\phantom{AA}R(\widetilde f)\phantom{AA}}{\longrightarrow} && R(c) \\ L(d) &&\underset{\phantom{AA}\widetilde f\phantom{AA}}{\longrightarrow}&& c }$$

By Prop. , this is equivalent to $(L(d), \eta_d)$ being the initial object in the comma category $c/R$, which in turn is equivalent to it being the limit of the identity functor on $c/R$ (by Example ). But this follows directly from the limit formulas (9) and (10).

Remark

(adjoint functor theorem)

Beware the subtle point in Prop. , that the comma category $c/F$ is in general not a small category (Def. ): It has typically “as many” objects as $\mathcal{C}$ has, and $\mathcal{C}$ is not assumed to be small (while of course it may happen to be). But typical categories, such as notably the category of sets (Example ) are generally guaranteed only to admit limits over small categories. For this reason, Prop. is rarely useful for finding an adjoint functor which is not already established to exist by other means.

But there are good sufficient conditions known, on top of the condition that $R$ preserves limits, which guarantee the existence of an adjoint functor, after all. This is the topic of the adjoint functor theorem (one of the rare instances of useful and non-trivial theorems in mathematics for which issues of set theoretic size play a crucial role for their statement and proof).

A very special but also very useful case of the adjoint functor theorem is the existence of adjoints of base change functors between categories of (enriched) presheaves via Kan extension. This we discuss as Prop. below. Since this is most conveniently phrased in terms of special limits/colimits called ends/coends (Def. below) we first discuss these.

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Ends and coends

For working with enriched categories (Def. ) , a certain shape of limits/colimits (Def. ) is particularly relevant: these are called ends and coends (Def. below). We here introduce these and then derive some of their basic properties, such as notably the expression for Kan extension in terms of (co-)ends (prop. below).

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Definition

((co)end)

Let $\mathcal{C}$ be a small $\mathcal{V}$-enriched category (Def. ). Let

$$F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

be an enriched functor (Def. ) out of the enriched product category of $\mathcal{C}$ with its opposite category (Def. ). Then:

1. The coend of $F$, denoted

   $$\overset{c \in \mathcal{C}}{\int} F(c,c) \;\in\; \mathcal{V} \,,$$

   is the coequalizer in $\mathcal{V}$ of the two actions encoded in $F$ via Example :

   $$\underset{c,d \in \mathcal{C}}{\coprod} \mathcal{C}(c,d) \otimes F(d,c) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} F(c,c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} F(c,c) \,.$$

2. The end of $F$, denoted

   $$\underset{c\in \mathcal{C}}{\int} F(c,c) \;\in\; \mathcal{V} \,,$$

   is the equalizer in $\mathcal{V}$ of the adjuncts of the two actions encoded in $F$ via example :

   $$\underset{c\in \mathcal{C}}{\int} F(c,c) \overset{\;\;equ\;\;}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} F(c,c) \underoverset {\underset{\underset{c,d}{\sqcup} \tilde \rho_{(c,d)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(d)}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c\in \mathcal{C}}{\prod} \big[ \mathcal{C}\left(c,d\right), \; F\left(c,d\right) \big] \,.$$

Example

For $\mathcal{V}$ a cosmos, let $G \in \mathcal{V}$ be a group object. There is the n the one-object $\mathcal{V}$-enriched category $\mathbf{B}G$ as in Example .

Then a $\mathcal{V}$-enriched functor

$$(X,\rho_l) \;\colon\; \mathbf{B}G \longrightarrow \mathcal{V}$$

is an object $X \coloneqq F(\ast) \; \in \mathcal{V}$ equipped with a morphism

$$\rho_l \;\colon\; G \otimes X \longrightarrow X$$

satisfying the action property. Hence this is equivalently an action of $G$ on $X$.

The opposite category (def. ) $(\mathbf{B}G)^{op}$ comes from the opposite group-object

$$(\mathbf{B}G)^{op} = \mathbf{B}(G^{op}) \,.$$

(The isomorphism $G \simeq G^{op}$ induces a canonical euqivalence of enriched categories $(\mathbf{B}G)^{op} \simeq \mathbf{B}G$.)

So an enriched functor

$$(Y,\rho_r) \;\colon\; (\mathbf{B} G)^{op} \longrightarrow \mathcal{V}$$

is equivalently a right action of $G$.

Therefore the coend of two such functors (def. ) coequalizes the relation

$$(x g,\;y) \sim (x,\; g y)$$

(where juxtaposition denotes left/right action) and is the quotient of the plain tensor product by the diagonal action of the group $G$:

$$\overset{\ast \in \mathbf{B}(G_+)}{\int} (Y,\rho_r)(\ast) \,\otimes\, (X,\rho_l)(\ast) \;\simeq\; Y \otimes_G X \,.$$

Example

(enriched natural transformations as ends)

Let $\mathcal{C}$ be a small enriched category (Def. ). For $F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{V}$ two enriched presheaves (Example ), the end (def. ) of the internal-hom-functor

$$[F(-),G(-)] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

is an object of $\mathcal{V}$ whose underlying set (Example ) is the set of enriched natural transformations $F \Rightarrow G$ (Def. )

$$Hom_{\mathcal{V}}\left(1, \left( \underset{c \in \mathcal{C}}{\int} \big[ F(c),G(c) \big] \right) \right) \;\simeq\; Hom_{[\mathcal{C}, \mathcal{V}]}(F,G) \,.$$

Proof

The underlying pointed set functor $Hom_{\mathcal{V}}(1,-)\colon \mathcal{V}\to Set$ preserves all limits, since hom-functors preserve limits (Prop. ). Therefore there is an equalizer diagram in Set of the form

$$Hom_{\mathcal{V}}\left(1, \left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \right) \overset{equ}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Hom_{\mathcal{V}}(F(c),G(c)) \underoverset {\underset{\underset{c,d}{\sqcup} U(\tilde \rho_{(c,d)}(c)) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(d))}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c,d\in \mathcal{C}}{\prod} Hom_{\mathcal{V}}( \mathcal{C}(c,d), [F(c),G(d)] ) \,,$$

where we used Example to identify underlying sets of internal homs with hom-sets.

Here the object in the middle is just the set of indexed sets of component morphisms $\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}$. The two parallel maps in the equalizer diagram take such a collection to the indexed set of composites (?) and (?). Hence that these two are equalized is precisely the condition that the indexed set of components constitutes an enriched natural transformation.

Conversely, example says that ends over bifunctors of the form $[F(-),G(-))]$ constitute hom-spaces between pointed topologically enriched functors:

Definition

(enriched presheaf category)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ be a small $\mathcal{V}$-enriched category (Def. ).

Then the $\mathcal{V}$-enriched presheaf category $[\mathcal{C}, \mathcal{V}]$ is $\mathcal{V}$-enriched functor category from $\mathcal{C}$ to $\mathcal{V}$, hence is the following $\mathcal{V}$-enriched category (Def. )

1. the objects are the $\mathcal{C}$-enriched functors $\mathcal{C} \overset{F}{\to}\mathcal{V}$ (Def. );

2. the hom-objects are the ends

   (11)$$[\mathcal{C}, \mathcal{V}](F,G) \;\coloneqq\; \int_{c\in \mathcal{C}} [F(c),G(c)]$$

3. the composition operation on these is defined to be the one induced by the composite maps

   $$\left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \otimes \left( \underset{c \in \mathcal{C}}{\int} [G(c),H(c)] \right) \overset{}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} [F(c),G(c)] \otimes [G(c),H(c)] \overset{(\circ_{F(c),G(c),H(c)})_{c\in \mathcal{C}}}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} [F(c),H(c)] \,,$$

   where the first morphism is degreewise given by projection out of the limits that defined the ends. This composite evidently equalizes the two relevant adjunct actions (as in the proof of example ) and hence defines a map into the end

   $$\left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \otimes \left( \underset{c \in \mathcal{C}}{\int} [G(c),H(c)] \right) \longrightarrow \underset{c\in \mathcal{C}}{\int} [F(c),H(c)] \,.$$

By Example , the underlying plain category (Example ) of this enriched functor category is the plain functor category of enriched functors from Example .

Proposition

(enriched Yoneda lemma)

For $\mathcal{V}$ a cosmos (Def. ) let $\mathcal{C}$ be a small enriched category (Def. ). For $F \colon \mathcal{C} \to \mathcal{V}$ an enriched presheaf (Example ) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism

$$[\mathcal{C}, \mathcal{V}](\mathcal{C}(c,-),\; F) \;\simeq\; F(c)$$

between the hom-object of the enriched functor category (Def. ), from the functor represented by $c$ to $F$, and the value of $F$ on $c$.

In terms of the ends (def. ) defining these hom-objects (11), this means that

$$\underset{d\in \mathcal{C}}{\int} [\mathcal{C}(c,d), F(d)] \;\simeq\; F(c) \,.$$

In this form the statement is also known as Yoneda reduction.

Now that natural transformations are expressed in terms of ends (example ), as is the enriched Yoneda lemma (prop. ), it is natural to consider the dual statement (Example ) involving coends:

Proposition

(enriched co-Yoneda lemma)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ be a small $\mathcal{V}$-enriched category (Def. ). For $F \colon \mathcal{C}\to \mathcal{V}$ an enriched presheaf (Def. ) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism

$$F(-) \simeq \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \,.$$

Moreover, the morphism that hence exhibits $F(c)$ as the coequalizer of the two morphisms in def. is componentwise the canonical action

$$\mathcal{C}(c,d) \otimes F(c) \longrightarrow F(d)$$

which is adjunct to the component map $\mathcal{C}(d,c) \to [F(c),F(d)]$ of the enriched functor $F$.

(e.g. MMSS 00, lemma 1.6)

Proof

By the definition of coends and the universal property of colimits, enriched natural transformations of the form

$$\left( \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \right) \longrightarrow G$$

are in natural bijection with systems of component morphisms

$$\mathcal{C}(c,d) \otimes F(c) \longrightarrow G(d)$$

which satisfy some compatibility conditions in their dependence on $c$ and $d$ (natural in $d$ and “extranatural” in $c$). By the internal hom adjunction, these are in natural bijection to systems of morphisms of the form

$$F(c) \longrightarrow [\mathcal{C}(c,d), G(d)]$$

satisfying the analogous compatibility conditions. By Example these are in natural bijection with systems of morphisms

$$F(c) \longrightarrow [\mathcal{C},\mathcal{V}](\mathcal{C}(c,-), G(-))$$

natural in $c$

By the enriched Yoneda lemma (Prop. ), these, finally, are in natural bijection with systems of morphisms

$$F(c) \longrightarrow G(c)$$

natural in $c$. Moreover, all these identifications are also natural in $G$. Therefore, in summary, this shows that there is a natural isomorphism

$$Hom_{[\mathcal{C},\mathcal{V}]} \left( \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \;,\; (-) \right) \;\simeq\; Hom_{[\mathcal{C},\mathcal{V}]} \left( F, (-) \right) \,.$$

With this, the ordinary Yoneda lemma (Prop. ) in the form of the Yoneda embedding of $[\mathcal{C},\mathcal{V}]$ implies the required isomorphism.

Example

(co-Yoneda lemma over Set)

Consider the co-Yoneda lemma (Prop. ) in the special case $\mathcal{V} =$ Set (Example ).

In this case the coequalizer in question is the set of equivalence classes of pairs

$$( c \overset{}{\to} c_0,\; x ) \;\; \in \mathcal{C}(c,c_0) \otimes F(c) \,,$$

where two such pairs

$$( c \overset{f}{\to} c_0,\; x \in F(c) ) \,,\;\;\;\; ( d \overset{g}{\to} c_0,\; y \in F(d) )$$

are regarded as equivalent if there exists

$$c \overset{\phi}{\to} d$$

such that

$$f = g \circ \phi \,, \;\;\;\;\;and\;\;\;\;\; y = \phi(x) \,.$$

(Because then the two pairs are the two images of the pair $(g,x)$ under the two morphisms being coequalized.)

But now considering the case that $d = c_0$ and $g = id_{c_0}$, so that $f = \phi$ shows that any pair

$$( c \overset{\phi}{\to} c_0, \; x \in F(c))$$

is identified, in the coequalizer, with the pair

$$(id_{c_0},\; \phi(x) \in F(c_0)) \,,$$

hence with $\phi(x)\in F(c_0)$.

As a conceptually important corollary we obtain:

Proposition

(category of presheaves is free co-completion)

For $\mathcal{C}$ a small category (Def. ), its Yoneda embedding $\mathcal{C} \overset{y}{\hookrightarrow} [\mathcal{C}^{op}, Set]$ (Prop. ) exhibits the category of presheaves $[\mathcal{C}^{op}, Set]$ (Example ) as the free co-completion of $\mathcal{C}$ under forming colimits (Def. ), in that it is a universal morphism, as in Def. but “up to natural isomorphism”, into a category with all colimits (by Example ) in the following sense:

1. for $\mathcal{D}$ any category with all colimits (Def. );

2. for $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ any functor;

there is a functor $\widetilde F \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow \mathcal{D}$, unique up to natural isomorphism such that

1. $\widetilde F$ preserves all colimits,

2. $\widetilde F$ extends $F$ through the Yoneda embedding, in that the following diagram commutes, up to natural isomorphism (Def. ):

$$\array{ && \mathcal{C} \\ & {}^{y}\swarrow &\swArrow& \searrow^{\mathrlap{F}} \\ \mathrlap{ \!\!\!\!\!\!\!\!\!\!\!\!\! [\mathcal{C}^{op}, Set] } && \underset{ \widetilde F }{\longrightarrow} && \mathcal{D} }$$

Hence when interpreting presheaves as generalized spaces, this says that “generalized spaces are precisely what is obtained from allowing arbitrary gluings of ordinary spaces”, see also Remark below.

Proof

The last condition says that $\widetilde F$ is fixed on representable presheaves by

(12)$$\widetilde F( y(c) ) \simeq F(c) \,.$$

and in fact naturally so:

(13)$$\array{ c_1&& \widetilde F( y(c_1) ) &\simeq& F(c_1) \\ {}^{\mathllap{f}}\big\downarrow && {}^{\mathllap{ F(y(f)) }}\big\downarrow && \big\downarrow^{\mathrlap{ F(f) }} \\ c_2 && \widetilde F (y(c_2)) &\simeq& F(c_2) }$$

But the co-Yoneda lemma (Prop. ) expresses every presheaf $\mathbf{X} \in [\mathcal{C}^{op}, Set]$ as a colimit of representable presheaves (in the special case of enrichment over $Set$, Example )

$$\mathbf{X} \;\simeq\; \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \,.$$

Since $\tilde F$ is required to preserve any colimit and hence these particular colimits, (12) implies that $\widetilde F$ is fixed to act, up to isomorphism, as

$$\begin{aligned} \widetilde F(\mathbf{X}) & = \widetilde F \left( \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \right) & \coloneqq \int^{c \in \mathcal{C}} F(c) \cdot \mathbf{X}(c) \;\;\;\;\in \mathcal{D} \end{aligned}$$

(where the colimit on the right is computed in $\mathcal{D}$!).

Remark

The statement of the co-Yoneda lemma in prop. is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):

For $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then

$$\int_{x \in X} \delta(x,x_0) f(x) = f(x_0) \,.$$

It is this analogy that gives the name to the following statement:

Proposition

(Fubini theorem for (co)-ends)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}_1, \mathcal{C}_2$ be two $\mathcal{V}$-enriched categories (Def. ) and

$$F \;\colon\; \left( \mathcal{C}_1\times\mathcal{C}_2 \right)^{op} \times (\mathcal{C}_1 \times\mathcal{C}_2) \longrightarrow \mathcal{V}$$

a $\mathcal{V}$-enriched functor (Def. ) from the product category with opposite categories (Def. ), as shown.

Then its end and coend (def. ) is equivalently formed consecutively over each variable, in either order:

$$\overset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_1}{\int} \overset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_2}{\int} \overset{c_1}{\int} F((c_1,c_2), (c_1,c_2))$$

and

$$\underset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_1}{\int} \underset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_2}{\int} \underset{c_1}{\int} F((c_1,c_2), (c_1,c_2)) \,.$$

Proof

Because limits commute with limits, and colimits commute with colimits.

Remark

(internal hom preserves ends)

Let $\mathcal{V}$ be a cosmos (Def. ). Since the internal hom-functor in $\mathcal{V}$ (Def. ) preserves limits in both variables (Prop. ), in particular it preserves ends (Def. ) in the second variable, and sends coends in the second variable to ends:

For all small $\mathcal{C}$-enriched categories, $\mathcal{V}$-enriched functors $F \;\colon\; \mathcal{C}^{op} \otimes \mathcal{C} \to \mathcal{V}$ (Def. ) and all objects $X \in \mathcal{V}$ we have natural isomorphisms

$$\left[ X , \int^{c \in \mathcal{C}} F(c,c) \right] \;\simeq\; \int^{c \in \mathcal{C}} \left[ X, F(c,c) \right]$$

and

$$\left[ \int_{c \in \mathcal{C}} F(c,c) , X \right] \;\simeq\; \int^{c \in \mathcal{C}} \left[ F(c,c), X \right] \,.$$

With this coend calculus in hand, there is an elegant proof of the defining universal property of the smash tensoring and powering enriched presheaves

Definition

(tensoring and powering of enriched presheaves)

Let $\mathcal{C}$ be a $\mathcal{V}$-enriched category, def. , with $[\mathcal{C}, \mathcal{V}]$ its functor category of enriched functors (Example ).

1. Define a functor

   $$(-)\cdot(-) \;\colon\; [\mathcal{C}, \mathcal{V}] \times \mathcal{V} \longrightarrow [\mathcal{C}, \mathcal{V}]$$

   by forming objectwise tensor products

   $$F \cdot X \;\colon\; c \mapsto F(c) \otimes X \,.$$

   This is called the tensoring of $[\mathcal{C}, \mathcal{V}]$ over $\mathcal{V}$.

2. Define a functor

   $$(-)^{(-)} \;\colon\; \mathcal{V}^{op} \times [\mathcal{C}, \mathcal{V}] \longrightarrow [\mathcal{C}, \mathcal{V}]$$

   by forming objectwise internal homs (Def. )

   $$F^X \;\colon\; c \mapsto [X,F(c)] \,.$$

   This is called the powering of $[\mathcal{C}, \mathcal{V}]$ over $\mathcal{V}$.

Proposition

(universal property of tensoring and powering of enriched presheaves)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ be a small $\mathcal{V}$-enriched category (Def. ), with $[\mathcal{C},\mathcal{V}]$ the corresponding enriched presheaf category.

Then there are natural isomorphisms

$$[\mathcal{C}, \mathcal{V}]( X \cdot K ,\, Y ) \;\simeq\; [K,\big( [\mathcal{C}, \mathcal{V}]\left( X, Y \right) \big)]$$

and

$$[\mathcal{C}, \mathcal{V}]\left( X,\, Y^K \right) \;\simeq\; [K, \big( [\mathcal{C}, \mathcal{C}](X,Y) \big) ]$$

for all $X,Y \in [\mathcal{C}, \mathcal{V}]$ and all $K \in \mathcal{C}$, where $(-)^K$ is the powering and $(-)\cdot K$ the tensoring from Def. .

In particular there is the composite natural isomorphism

$$[\mathcal{C}, \mathcal{V}](X \cdot K, Y) \;\simeq\; [\mathcal{C}, \mathcal{V}]\left( X, Y^K \right)$$

exhibiting a pair of adjoint functors

$$[\mathcal{C}, \mathcal{V}] \underoverset {\underset{(-)^K}{\longrightarrow}} {\overset{(-) \cdot K}{\longleftarrow}} {\bot} [\mathcal{C}, \mathcal{V}] \,.$$

Proof

Via the end-expression for $[\mathcal{C}, \mathcal{V}](-,-)$ from Example , and the fact (remark ) that the internal hom-functor ends in the second variable, this reduces to the fact that $[-,-]$ is the internal hom in the closed monoidal category $\mathcal{V}$ (Example ) and hence satisfies the internal tensor/hom-adjunction isomorphism (prop. ):

$$\begin{aligned} [\mathcal{C}, \mathcal{V}](X \cdot K, Y) & = \underset{c}{\int} [ (X \cdot K)(c), Y(c) ] \\ & \simeq \underset{c}{\int} [X(c) \otimes K, Y(x)] \\ & \simeq \underset{c}{\int} [K,[X(c), Y(c)]] \\ & \simeq [K, \underset{c}{\int} [X(c),Y(c)]] \\ & = [K,\left( [\mathcal{C},\mathcal{V}](X,Y)\right)] \end{aligned}$$

and

$$\begin{aligned} [\mathcal{C}, \mathcal{V}](X, Y^K) & = \underset{c}{\int} [X(c), Y^K(c)] \\ & \simeq \underset{c}{\int} [ X(c), [K,Y(c)] ] \\ & \simeq \underset{c}{\int} [ X(c) \otimes K, Y(c) ] \\ & \simeq \underset{c}{\int} [K, [X(c),Y(c)]] \\ & \simeq [K, \underset{c}{\int} [X(c),Y(c)] ] \\ & \simeq [K, [\mathcal{C}, \mathcal{V}](X,Y] \,. \end{aligned}$$

$\,$

Tensoring and cotensoring

We make explicit the general concept of which Prpp. provides a key class of examples:

$\,$

Definition

(tensoring and cotensoring)

For $\mathcal{V}$ a cosmos (Def. ) let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ). Recall the enriched hom-functors (Example )

$$\mathcal{C}(-,-) \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

and (via Example )

$$\mathcal{V}(-,-) = [-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{V} \longrightarrow \mathcal{V} \,.$$

1. A powering (or cotensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a functor (Def. )

      $$[-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$$

   2. for each $v \in \mathcal{V}$ a natural isomorphism (Def. ) of the form

      (14)$$\mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2])$$

2. A copowering (or tensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a functor (Def. )

      $$(-)\otimes(-) \;\colon\; \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}$$

   2. for each $v \in \mathcal{V}$ a natural isomorphism (Def. ) of the form

      (15)$$\mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) )$$

If $\mathcal{C}$ is equipped with a (co-)powering it is called (co-)powered over $\mathcal{V}$.

Proposition

(tensoring left adjoint to cotensoring)

For $\mathcal{V}$ a cosmos (Def. ) let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ).

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ (Def. ), then for fixed $v \in \mathcal{V}$ the operations of tensoring with $v$ and of cotensoring with $\mathcal{V}$ form a pair of adjoint functors (Def. )

$$\mathcal{C} \underoverset {\underset{ [v,-] }{\longrightarrow}} {\overset{ v \otimes (-) }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}$$

Proof

The hom-isomorphism (?) characterizing the pair of adjoint functors is provided by the composition of the natural isomorphisms (14) and (15):

$$\array{ \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2]) }$$

Proposition

(in tensored and cotensored categories initial/terminal objects are enriched initial/terminal)

For $\mathcal{V}$ a cosmos (Def. ) let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ).

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ (Def. ) then

1. an initial object $\emptyset$ (Def. ) of the underlying category of $\mathcal{C}$ (Example ) is also enriched initial, in that the hom-object out of it is the terminal object $\ast$ of $\mathcal{V}$

   $$\mathcal{C}(\emptyset, c) \;\simeq\; \ast$$

2. a terminal object $\ast$ (Def. ) of the underlying category of $\mathcal{C}$ (Example ) is also enriched terminal, in that the hom-object into it is the terminal object of $\mathcal{V}$:

   $$\mathcal{C}(c, \ast) \;\simeq\; \ast$$

Proof

We discuss the first claim, the second is formally dual.

By prop. , tensoring is a left adjoint. Since left adjoints preserve colimits (Prop. ), and since an initial object is the colimit over the empty diagram (Example ), it follows that

$$v \otimes \emptyset \;\simeq\; \emptyset$$

for all $v \in \mathcal{V}$, in particular for $\emptyset \in \mathcal{V}$. Therefore the natural isomorphism (15) implies for all $v \in \mathcal{V}$ that

$$\mathcal{C}(\emptyset, c) \;\simeq\; \mathcal{C}( \emptyset \otimes \emptyset, c ) \;\simeq\; \mathcal{V}( \emptyset, \mathcal{C}(\emptyset, c) ) \;\simeq\; \ast$$

where in the last step we used that the internal hom $\mathcal{V}(-,-) = [-,-]$ in $\mathcal{V}$ sends colimits in its first argument to limits (Prop. ) and used that a terminal object is the limit over the empty diagram (Example ).

$\,$

Kan extensions

Proposition

(Kan extension)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}, \mathcal{D}$ be small $\mathcal{V}$-enriched categories (Def. ) and let

$$p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

be a $\mathcal{V}$-enriched functor (Def. ). Then precomposition with $p$ constitutes a functor between the corresponding $\mathcal{V}$-enriched presheaf categories (Def. )

(16)$$p^\ast \;\colon\; \array{ [\mathcal{D}, \mathcal{V}] &\longrightarrow& [\mathcal{C}, \mathcal{V}] \\ G &\mapsto& G \circ p }$$

1. This enriched functor $p^\ast$ (16) has an enriched left adjoint $Lan_p$ (Def. ), called left Kan extension along $p$

   $$[\mathcal{D}, \mathcal{V}] \underoverset {\underset{p^\ast}{\longrightarrow}} {\overset{Lan_p }{\longleftarrow}} {\bot} [\mathcal{C}, \mathcal{V}]$$

   which is given objectwise by the coend (def. ):

   (17)$$(Lan_p F) \;\colon\; d \;\mapsto \; \overset{c\in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \otimes F(c) \,.$$

2. The enriched functor $p^\ast$ (16) has an enriched right adjoint $Ran_p$ (Def. ), called right Kan extension along $p$

   $$[\mathcal{C}, \mathcal{V}] \underoverset {\underset{Ran_p}{\longrightarrow}} {\overset{p^\ast}{\longleftarrow}} {\bot} [\mathcal{D}, \mathcal{V}]$$

   which is given objectwise by the end (def. ):

   (18)$$(Ran_p F) \;\colon\; d \;\mapsto \; \underset{c\in \mathcal{C}}{\int} [\mathcal{D}(d,p(c)), F(c)] \,.$$

In summary, this means that the enriched functor

$$\mathcal{C} \overset{p}{\longrightarrow} \mathcal{D}$$

induces, via Kan extension, an adjoint triple (Remark ) of enriched functors

(19)$$Lan_p \;\dashv\; p^\ast \;\dashv\; Ran_p \;\colon\; [\mathcal{C},\mathcal{V}] \leftrightarrow [\mathcal{D}, \mathcal{V}] \,.$$

Proof

Use the expression of enriched natural transformations in terms of coends (example and def. ), then use the respect of $[-,-]$ for ends/coends (remark ), use the internal-hom adjunction (?), use the Fubini theorem (prop. ) and finally use Yoneda reduction (prop. ) to obtain a sequence of natural isomorphisms as follows:

$$\begin{aligned} [\mathcal{D}, \mathcal{V}]( Lan_p F, \, G ) & = \underset{d \in \mathcal{D}}{\int} [ (Lan_p F)(d), \, G(d) ] \\ & = \underset{d\in \mathcal{D}}{\int} \left[ \overset{c \in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \otimes F(c) ,\; G(d) \right] \\ &\simeq \underset{d \in \mathcal{D}}{\int} \underset{c \in \mathcal{C}}{\int} [ \mathcal{D}(p(c),d) \otimes F(c) \,,\; G(d) ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} \underset{d\in \mathcal{D}}{\int} [F(c), [ \mathcal{D}(p(c),d) , \, G(d) ] ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} [F(c), \underset{d\in \mathcal{D}}{\int} [ \mathcal{D}(p(c),d) , \, G(d) ] ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} [ F(c), G(p(c)) ] \\ & = [\mathcal{C}, \mathcal{V}](F,p^\ast G) \end{aligned} \,.$$

and similarly:

$$\begin{aligned} [\mathcal{D}, \mathcal{V}]( G,\, Ran_p F ) & \simeq \underset{d \in \mathcal{D}}{\int} [ G(d) ,\, (Ran_p F)(d), \, ] \\ & \simeq \underset{d \in \mathcal{D}}{\int} \left[ G(d) ,\, \underset{c\in \mathcal{C}}{\int} [\mathcal{D}(d,p(c)), F(c)] \right] \\ & \simeq \underset{d \in \mathcal{D}}{\int} \underset{c\in \mathcal{C}}{\int} \left[ G(d) \otimes \mathcal{D}(d,p(c)),\, F(c) \right] \\ & \simeq \underset{c\in \mathcal{C}}{\int} \left[ \overset{d \in \mathcal{D}}{\int} G(d) \otimes \mathcal{D}(d,p(c)),\, F(c) \right] \\ & \simeq \underset{c \in \mathcal{D}}{\int} \left[ G(p(c)),\, F(c) \right] \\ & \simeq [\mathcal{C}, \mathcal{V}]( p^\ast G , F ) \end{aligned}$$

Example

(coend formula for left Kan extension of ordinary presheaves)

Consider the cosmos to be $\mathcal{V} =$ Set, via Example , so that small $\mathcal{V}$-enriched categories (Def. ) are just a plain small category (Def. ) by Example , and $\mathcal{V}$-enriched presheaves (Example ) are just plain presheaves (Example ).

Then for any plain functor (Def. )

$$\mathcal{C}^{op} \overset{\phantom{AA} p \phantom{AA}}{\longrightarrow} (\mathcal{C}')^{op}$$

the general formula (17) for left Kan extension

$$[\mathcal{C}^{op},Set] \overset{Lan_p}{\longrightarrow} [(\mathcal{C}')^{op}, Set]$$

is

$$(Lan_p F)(c') \simeq \int^{c \in C} C'(c', p(c)) \times F(c) \,.$$

Using here the Yoneda lemma (Prop. ) to rewrite $F(c) \simeq Hom_{PSh(C)}(c,F)$, this is

$$(Lan_p F)(c') \simeq \int^{c \in C} Hom_{C'}(c', p(c)) \times Hom_{PSh(C)}(c,F) \,.$$

Hence this coend-set consists of equivalence classes of pairs of morphisms

$$(c' \to p(c), c \to F)$$

where two such are regarded as equivalent whenever there is $f \colon c'_1 \to c'_2$ such that

$$\array{ && c' \\ & \swarrow && \searrow \\ p(c_1) && \stackrel{p(f)}{\longrightarrow} && p(c_2) \\ c_1 && \stackrel{f}{\longrightarrow} && c_2 \\ & \searrow && \swarrow \\ && F } \,.$$

This is particularly suggestive when $p$ is a full subcategory inclusion (Def. ). For in that case we may imagine that a representative pair $(c' \to p(c), c \to F)$ is a stand-in for the actual pullback of elements of $F$ along the would-be composite “$c'\to c \to F$”, only that this composite need not be defined. But the above equivalence relation is precisely that under which this composite would be invariant.

Further properties

We collect here further key properties of the various universal constructions considered above.

$\,$

Proposition

(left Kan extension preserves representable functors)

For $\mathcal{V}$ a cosmos (Def. ), let

$$\mathcal{C} \overset{p}{\longrightarrow} \mathcal{D}$$

be a $\mathcal{V}$-enriched functor (Def. ) between small $\mathcal{V}$-enriched categories (Def. ).

Then the left Kan extension $Lan_p$ (Prop. ) takes representable enriched presheaves $\mathcal{C}(c,-) \;\colon\; \mathcal{C} \to \mathcal{V}$ to their image under $p$:

$$Lan_p \mathcal{C}(c, -) \;\simeq\; \mathcal{D}(p(c), -)$$

for all $c \in C$.

Proof

By the coend formula (17) we have, naturally in $d' \in \mathcal{D}$, the expression

$$\begin{aligned} Lan_p \mathcal{C}(c,-) \;\colon\; d' \mapsto & \int^{c' \in C} \mathcal{D}(p(c'), d') \otimes \mathcal{C}(c,-)(c') \\ & \simeq \int^{c' \in C} \mathcal{D}(p(c'), d') \otimes \mathcal{C}(c,c') \\ & \simeq \mathcal{D}(p(c), d') \end{aligned} \,,$$

where the last step is the co-Yoneda lemma (Prop. ).

Example

(Kan extension of adjoint pair is adjoint quadruple)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$, $\mathcal{D}$ be two small $\mathcal{V}$-enriched categories (Def. ) and let

$$\mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}$$

be a $\mathcal{V}$-enriched adjunction (Def. ). Then there are $\mathcal{V}$-enriched natural isomorphisms (Def. )

$$(q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]$$

$$(p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]$$

between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other (Def. ).

By essential uniqueness of adjoint functors, this means that the two adjoint triples (Remark ) given by Kan extension (19) of $q$ and $p$

$$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }$$

merge into an adjoint quadruple (Remark )

$$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$$

Proof

For every enriched presheaf $F \;\colon\; \mathcal{C}^{op} \to \mathcal{V}$ we have a sequence of $\mathcal{V}$-enriched natural isomorphism as follows

$$\begin{aligned} (Lan_{p^{op}} F)(d) & \simeq \int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c) \\ & \simeq \int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c) \\ & \simeq F(q(d)) \\ & = \left( (q^{op})^\ast F\right) (d) \,. \end{aligned}$$

Here the first step is the coend-formula for left Kan extension (Prop. ), the second step if the enriched adjunction-isomorphism (?) for $q \dashv p$ and the third step is the co-Yoneda lemma.

This shows the first statement, which, by essential uniqueness of adjoints, implies the following statements.

Proposition

(left Kan extension along fully faithful functor is fully faithful)

For $\mathcal{V}$ a cosmos (Def. ), let

$$\mathcal{C} \overset{\phantom{AA} p \phantom{AA}}{\hookrightarrow} \mathcal{D}$$

be a fully faithful $\mathcal{V}$-enriched functor (Def. ) between small $\mathcal{V}$-enriched categories (Def. ).

Then for all $c \in \mathcal{C}$

$$p^* (Lan_p c) \simeq c$$

and in fact the $(Lan_F \dashv F^*)$-unit of an adjunction is a natural isomorphism

$$Id \stackrel{\simeq}{\to} p^* \circ Lan_{p} \,.$$

hence, by Prop. ,

$$[\mathcal{C}^{op}, Set] \overset{\phantom{AA} Lan_p \phantom{AA}}{\hookrightarrow} [\mathcal{D}^{op}, Set]$$

is a fully faithful functor.

Proof

By the coend formula (17) we have, naturally in $d' \in \mathcal{D}$, the left Kan extension of any $F \;\colon\; \mathcal{C} \to \mathcal{V}$ on the image of $p$ is

$$\begin{aligned} Lan_p F \;\colon\; p(c) \mapsto & \int^{c' \in C} \mathcal{D}(p(c'), p(c)) \cdot F(c') \\ & \simeq \int^{c' \in C} \mathcal{C}(c', c) \cdot F(c') \\ & \simeq F(c) \end{aligned} \,,$$

where in the second step we used the assumption of fully faithfulness of $p$ and in the last step we used the co-Yoneda lemma (Prop. ).

Lemma

(colimit of representable is singleton)

Let $\mathcal{C}$ be a small category (Def. ). Then the colimit of a representable presheaf (Def. ), regarded as a functor

$$y(c) \;\colon\; \mathcal{C}^{op} \longrightarrow Set$$

is the singleton set.

(20)$$\underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) \;\simeq\; \ast \,.$$

Proof

One way to see this is to regard the colimit as the left Kan extension (Prop. ) along the unique functor $\mathcal{C}^{op} \overset{p}{\to} \ast$ to the terminal category (Def. ). By the formula (17) this is

$$\begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) & \simeq \int^{c_1 \in \mathcal{C}} \underset{const_\ast(c_1)}{\underbrace{\ast(-,p(c_1))}} \times y(c)(c_1) \\ & \simeq \int^{c_1 \in \mathcal{C}} const_\ast(c_1) \times \mathcal{C}(c_1,c) \\ & \simeq const_\ast(c) \\ & \simeq \ast \end{aligned}$$

where we made explicit the constant functor $const_\ast \;\colon\; \mathcal{C} \to Set$, constant on the singleton set $\ast$, and then applied the co-Yoneda lemma (Prop. ).

Proposition

(categories with finite products are cosifted

Let $\mathcal{C}$ be a small category (Def. ) which has finite products. Then $\mathcal{C}$ is a cosifted category, equivalently its opposite category $\mathcal{C}^{op}$ is a sifted category, equivalently colimits over $\mathcal{C}^{op}$ with values in Set are sifted colimits, equivalently colimits over $\mathcal{C}^{op}$ with values in Set commute with finite products, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to functors on the opposite category of $\mathcal{C}$ (hence two presheaves on $\mathcal{C}$, Example ) we have a natural isomorphism (Def. )

$$\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right)$$

between the colimit of their Cartesian product and the Cartesian product of their separate colimits.

Proof

First observe that for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ two presheaves, their Cartesian product is a colimit over presheaves represented by Cartesian products in $\mathcal{C}$. Explicity, using coend-notation, we have:

(21)$$\mathbf{X} \times \mathbf{Y} \;\simeq\; \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \,,$$

where $y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set]$ denotes the Yoneda embedding.

This is due to the following sequence of natural isomorphisms:

$$\begin{aligned} (\mathbf{X} \times \mathbf{Y})(c) & \simeq \left( \int^{c_1 \in \mathcal{C}} \mathcal{C}(c,c_1) \times \mathbf{X}(c_1) \right) \times \left( \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_2) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \underset{ \simeq \mathcal{c}(c, c_1 \times c_2) }{ \underbrace{ \mathcal{C}(c,c_1) \times \mathcal{C}(c,c_2) }} \times \left( \mathbf{X}(c_1) \times \mathbf{X}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{X}(c_2) \,, \end{aligned}$$

where the first step expands out both presheaves as colimits of representables separately, via the co-Yoneda lemma (Prop. ), the second step uses that the Cartesian product of presheaves is a two-variable left adjoint (by the symmetric closed monoidal structure on presheaves) and as such preserves colimits (in particular coends) in each variable separately (Prop. ), and under the brace we use the defining universal property of the Cartesian products, assumed to exist in $\mathcal{C}$.

With this, we have the following sequence of natural isomorphisms:

$$\begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) & \simeq \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \underset{ \simeq \ast }{ \underbrace{ \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c_1 \times c_2) }} \right) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \left( \int^{c_1\in \mathcal{C}} \mathbf{X}(c_1) \right) \times \left( \int^{c_2\in \mathcal{C}} \mathbf{Y}(c_2) \right) \\ & \simeq \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \end{aligned}$$

Here the first step is (21), the second uses that colimits commute with colimits (Prop. ), the third uses again that the Cartesian product respects colimits in each variable separately, the fourth is by Lemma , the last step is again the respect for colimits of the Cartesian product in each variable separately.