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\begin{document}

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\section*{adjoint (infinity,1)-functor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_homequivalences}{In terms of hom-equivalences}\dotfill \pageref*{in_terms_of_homequivalences} \linebreak
\noindent\hyperlink{InTermsOfCographsHeteromorphisms}{In terms of cographs / heteromorphisms}\dotfill \pageref*{InTermsOfCographsHeteromorphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{uniqueness_of_adjoints}{Uniqueness of adjoints}\dotfill \pageref*{uniqueness_of_adjoints} \linebreak
\noindent\hyperlink{PresOfLims}{Preservation of limits and colimits}\dotfill \pageref*{PresOfLims} \linebreak
\noindent\hyperlink{OnHomotopyCat}{Adjunctions on homotopy categories}\dotfill \pageref*{OnHomotopyCat} \linebreak
\noindent\hyperlink{FullAndFaithfulAdjoints}{Full and faithful adjoints}\dotfill \pageref*{FullAndFaithfulAdjoints} \linebreak
\noindent\hyperlink{OnSlices}{On over-$(\infty,1)$-categories}\dotfill \pageref*{OnSlices} \linebreak
\noindent\hyperlink{UniversalArrows}{In terms of universal arrows}\dotfill \pageref*{UniversalArrows} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{quillen_adjunctions}{Quillen adjunctions}\dotfill \pageref*{quillen_adjunctions} \linebreak
\noindent\hyperlink{SimplicialAndDerived}{Simplicial and derived adjunctions}\dotfill \pageref*{SimplicialAndDerived} \linebreak
\noindent\hyperlink{localizations}{Localizations}\dotfill \pageref*{localizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of adjunction between two [[(∞,1)-functors]] generalizes the notion of [[adjoint functors]] from [[category theory]] to [[(infinity,1)-category|(∞,1)-category theory]].

There are many equivalent definitions of the ordinary notion of [[adjoint functor]]. Some of them have more evident generalizations to some parts of [[higher category theory]] than others.

\begin{itemize}%
\item One definition of ordinary adjoint functors says that a pair of functors $C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ is an adjunction if there is a [[natural transformation|natural isomorphism]]

\begin{displaymath}
Hom_C(L(-),(-) \simeq Hom_D(-,R(-))
  \,.
\end{displaymath}
The analog of this definition makes sense very generally in [[(∞,1)-category theory]], where $Hom_C(-,-) : C^{op} \times C \to \infty Grpd$ is the $(\infty,1)$-categorical hom-object.


\item One other characterization of adjoint functors in terms of their [[cograph of a functor|cographs]]/[[heteromorphisms]]: the [[Cartesian fibrations]] to which the . At [[cograph of a functor]] it is discussed how two functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if the cograph of $L$ coincides with the cograph of $R$ up to the obvious reversal of arrows



\end{itemize}
\begin{displaymath}
(L \dashv R) \Leftrightarrow
  (cograph(L) \simeq cograph(R^{op})^{op})
  \,.
\end{displaymath}
Using the [[(∞,1)-Grothendieck construction]] the notion of cograph of a functor has an evident generalization to $(\infty,1)$-categories.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_terms_of_homequivalences}{}\subsubsection*{{In terms of hom-equivalences}}\label{in_terms_of_homequivalences}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(in terms of hom equivalence induced by unit map)}

A pair of [[(∞,1)-functors]]

\begin{displaymath}
C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} 
  D
\end{displaymath}
is an adjunction, if there exists a \emph{unit transformation} $\epsilon : Id_D \to R \circ L$ -- a morphism in the [[(∞,1)-category of (∞,1)-functors]] $Func(D,D)$ -- such that for all $d \in D$ and $c \in C$ the induced morphism

\begin{displaymath}
Hom_C(L(d),c)
   \stackrel{R_{L(d), c}}{\to}  
  Hom_D(R(L(d)), R(c))
  \stackrel{Hom_D(\epsilon, R(c))}{\to}
  Hom_D(d,R(c))
\end{displaymath}
is an [[equivalence of ∞-groupoids]].

\end{defn}
In terms of the concrete incarnation of the notion of $(\infty,1)$-category by the notion of [[quasi-category]], we have that $Hom_C(L(d),c)$ and $Hom_D(d,R(c))$ are incarnated as [[hom-object in a quasi-category|hom-objects in quasi-categories]], which are [[Kan complexes]], and the above equivalence is a [[homotopy equivalence]] of Kan complexes.

In this form this definition appears as [[Higher Topos Theory|HTT, def. 5.2.2.7]].

\hypertarget{InTermsOfCographsHeteromorphisms}{}\subsubsection*{{In terms of cographs / heteromorphisms}}\label{InTermsOfCographsHeteromorphisms}

We discuss here the quasi-category theoretic analog of \emph{\href{cograph+of+a+functor#AdjointFunctorsInTermsOfCographs}{Adjoint functors in terms of cographs}} ([[heteromorphisms]]).

We make use here of the explicit realization of the [[(∞,1)-Grothendieck construction]] in its incarnation for [[quasi-categories]]: here an [[(∞,1)-functors]] $L : D \to C$ may be regarded as a map $\Delta[1]^{op} \to$ [[(∞,1)Cat]], which corresponds under the Grothendieck construction to a [[Cartesian fibration]] of [[simplicial sets]] $coGraph(L) \to \Delta[1]$.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(in terms of Cartesian/coCartesian fibrations)}

Let $C$ and $D$ be [[quasi-categories]]. An \textbf{adjunction} between $C$ and $D$ is

\begin{itemize}%
\item a morphism $K \to \Delta[1]$ of [[simplicial sets]], which is both a [[Cartesian fibration]] as well as a coCartesian fibration.


\item together with [[equivalence of quasi-categories]] $C \stackrel{\simeq}{\to} K_{\{0\}}$ and $D \stackrel{\simeq}{\to} K_{\{1\}}$.



\end{itemize}
Two [[(∞,1)-functors]] $L : C \to D$ and $R : D \to C$ are called \textbf{adjoint} -- with $L$ \emph{left adjoint} to $R$ and $R$ \emph{right adjoint} to $L$ if

\begin{itemize}%
\item there exists an adjunction $K \to I$ in the above sense


\item and $L$ and $K$ are the  the Cartesian fibation $p \colon K \to \Delta[1]$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta[1]^{op}$, respectively.



\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The two different definition above are indeed equivalent:

\begin{prop}
\label{}\hypertarget{}{}
For $C$ and $D$ [[quasi-categories]], the two definitions of adjunction, in terms of Hom-equivalence induced by unit maps and in terms of Cartesian/coCartesian fibrations are equivalent.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 5.2.2.8]].

First we discuss how to produce the unit for an adjunction from the data of a correspondence $K \to \Delta[1]$ that encodes an $\infty$-adjunction $(f \dashv g)$.

For that, define a morphism $F' : \Lambda[2]_2 \times C \to K$ as follows:

\begin{itemize}%
\item on $\{0,2\}$ it is the morphism $F : C \times \Delta[1] \to K$ that exhibits $f$ as associated to $K$, being $Id_C$ on $C \times \{0\}$ and $f$ on $C \times \{2\}$;


\item on $\{1,2\}$ it is the morphism $C \times \Delta[1] \stackrel{f \times Id}{\to} D \times \Delta[1] \stackrel{G}{\to} K$, where $G$ is the morphism that exhibits $g$ as associated to $K$;



\end{itemize}
Now observe that $F'$ in particular sends $\{1,2\}$ to [[Cartesian morphism]]s in $K$ (by definition of functor associated to $K$). By one of the equivalent characterizations of [[Cartesian morphism]]s, this means that the lift in the diagram

\begin{displaymath}
\itexarray{
    \Lambda[2]_2 &\stackrel{F'}{\to}& K
    \\
    \downarrow &{}^{F''}\nearrow& \downarrow
    \\
    \Delta[2] \times C &\to & \Delta[1] 
  }
\end{displaymath}
exists. This defines a morphism $C \times \{0,1\} \to K$ whose components may be regarded as forming a [[natural transformation]] $u : d_C \to g \circ f$.

To show that this is indeed a unit transformation, we need to show that the maps of [[hom-object in a quasi-category]] for all $c \in C$ and $d \in D$

\begin{displaymath}
Hom_D(f(f), d) \to Hom_C(g(f(c)), g(d)) \to Hom_C(c, g(d))
\end{displaymath}
is an equivalence, hence an isomorphism in the [[homotopy category]]. Once checks that this fits into a commuting diagram

\begin{displaymath}
\itexarray{
    Hom_D(f(c), d) &\to& Hom_C(g(f(c)), g(d)) &\to& Hom_C(c, g(d))
    \\
    \downarrow &&&& \downarrow
    \\
    Hom_K(C,D) &&=&& Hom_K(C,D)
  }
  \,.
\end{displaymath}
For illustration, chasing a morphism $f(c) \to d$ through this diagram yields

\begin{displaymath}
\itexarray{
    (f(c) \to d) &\mapsto& (g(f(c)) \to g(d)) &\mapsto& 
    (c \to g(f(c)) \to g(d))
    \\
    \downarrow && && \downarrow
    \\
    (c \to g(f(c)) \to f(c) \to d)
    &&=&&
    (c \to g(f(c)) \to g(d) \to d)
  }
 \,,
\end{displaymath}
where on the left we precomposed with the Cartesian morphism

\begin{displaymath}
\itexarray{
    && g(f(c))
    \\
    & \nearrow &\Downarrow^{\simeq}& \searrow
    \\
    c &&\to&& f(c)
  }
\end{displaymath}
given by $F''|_{c} : \Delta[2] \to K$, by \ldots{}

\end{proof}
\hypertarget{uniqueness_of_adjoints}{}\subsubsection*{{Uniqueness of adjoints}}\label{uniqueness_of_adjoints}

The adjoint of a functor is, if it exists, essentially unique:

\begin{prop}
\label{}\hypertarget{}{}
If the $(\infty,1)$-functor between quasi-categories $L : D \to C$ admits a right adjoint $R : C \to D$, then this is unique up to homotopy.

Moreover, even the choice of homotopy is unique, up to ever higher homotopy, i.e. the collection of all right adjoints to $L$ forms a [[contractible]] [[∞-groupoid]], in the following sense:

Let $Func^L(C,D), Func^R(C,D) \subset Func(C,D)$ be the full sub-quasi-categories on the [[(∞,1)-category of (∞,1)-functors]] between $C$ and $D$ on those functors that are left adjoint and those that are right adjoints, respectively. Then there is a canonical [[equivalence of quasi-categories]]

\begin{displaymath}
Func^L(C,D) \stackrel{\simeq}{\to}
  Func^R(D,C)^{op}
\end{displaymath}
(to the [[opposite quasi-category]]), which takes every left adjoint functor to a corresponding right adjoint.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 5.2.1.3]] (also remark 5.2.2.2), and [[Higher Topos Theory|HTT, prop. 5.2.6.2]].

\end{proof}
\hypertarget{PresOfLims}{}\subsubsection*{{Preservation of limits and colimits}}\label{PresOfLims}

Recall that for $(L \dashv R)$ an ordinary pair of [[adjoint functor]]s, the fact that $L$ preserves [[colimit]]s (and that $R$ preserves [[limit]]s) is a formal consequence of

\begin{enumerate}%
\item the hom-isomorphism $Hom_C(L(-),-) \simeq Hom_D(-,R(-))$;


\item the fact that $Hom_C(-,-) : C^{op} \times C \to Set$ preserves all limits in both arguments;


\item the [[Yoneda lemma]], which says that two objects are isomorphic if all homs out of (into them) are.



\end{enumerate}
Using this one computes for all $c \in C$ and diagram $d : I \to D$

\begin{displaymath}
\begin{aligned}
  Hom_C(L(\lim_{\to} d_i), c) & \simeq 
  Hom_D(\lim_\to d_i, R(c))
  \\
  & \simeq 
  \lim_{\leftarrow} Hom_D(d_i, R(c))
  \\
  & \simeq \lim_{\leftarrow} Hom_C(L(d_i), c)
  \\
  & \simeq Hom_C(\lim_{\to} L(d_i), c)
  \,,
\end{aligned}
\end{displaymath}
which implies that $L(\lim_\to d_i) \simeq \lim_\to L(d_i)$.

Now to see this in $(\infty,1)$-category theory (\ldots{}) HTT Proposition 5.2.3.5

\hypertarget{OnHomotopyCat}{}\subsubsection*{{Adjunctions on homotopy categories}}\label{OnHomotopyCat}

\begin{prop}
\label{}\hypertarget{}{}
For $(L \dashv R) : C \stackrel{\leftarrow}{\to} D$ an $(\infty,1)$-adjunction, its image under decategorifying to [[homotopy category of an (infinity,1)-category|homotopy categories]] is a pair of ordinary [[adjoint functor]]s

\begin{displaymath}
(Ho(L) \dashv Ho(R)) : Ho(C) \stackrel{\leftarrow}{\to} Ho(D)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 5.2.2.9]].

This follows from that fact that for $\epsilon : Id_C \to R \circ L$ a unit of the $(\infty,1)$-adjunction, its image $Ho(\epsilon)$ is a unit for an ordinary adjunction.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The converse statement is in general false.

One way to find that an ordinary adjunction of homotopy categories lifts to an $(\infty,1)$-adjunction is to exhibit it as a [[Quillen adjunction]] between [[simplicial model category]]-structures. This is discussed in the Examples-section \hyperlink{SimplicialAndDerived}{Simplicial and derived adjunction} below.

\end{remark}
\hypertarget{FullAndFaithfulAdjoints}{}\subsubsection*{{Full and faithful adjoints}}\label{FullAndFaithfulAdjoints}

As for ordinary [[adjoint functors]] we have the following relations between full and faithful adjoints and idempotent monads.

\begin{prop}
\label{}\hypertarget{}{}
Given an $(\infty,1)$-adjunction $(L \dashv R) : C \to  D$

\begin{itemize}%
\item $R$ is a [[full and faithful (∞,1)-functor]] precisely is the counit $L R \stackrel{}{\to} Id$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functor]]

In this case $C$ is a [[reflective (∞,1)-subcategory]] of $D$.


\item $L$ is a [[full and faithful (∞,1)-functor]] precisely is the unit $Id \to L R$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functors]].



\end{itemize}
\end{prop}
\hyperlink{Lurie}{Lurie, prop. 5.2.7.4}, See also top of p. 308.

\hypertarget{OnSlices}{}\subsubsection*{{On over-$(\infty,1)$-categories}}\label{OnSlices}

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
     C
\end{displaymath}
be a pair of adjoint $(\infty,1)$-functors where the $(\infty,1)$-category $C$ has all [[(∞,1)-pullback]]s.

Then for every object $X \in C$ there is induced a pair of adjoint $(\infty,1)$-functors between the [[over-(∞,1)-categories]]

\begin{displaymath}
(L/X \dashv R/X) 
   : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}}
     C/X
\end{displaymath}
where

\begin{itemize}%
\item $L/X$ is the evident induced functor;


\item $R/X$ is the composite

\begin{displaymath}
R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to}
   C/X
\end{displaymath}
of the evident functor induced by $R$ with the [[(∞,1)-pullback]] along the $(L \dashv R)$-[[unit of an adjunction|unit]] at $X$.



\end{itemize}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.2.5.1]].

\hypertarget{UniversalArrows}{}\subsubsection*{{In terms of universal arrows}}\label{UniversalArrows}

\begin{prop}
\label{UnivArr}\hypertarget{UnivArr}{}
An $(\infty,1)$-functor $G:D\to C$ admits a left adjoint if and only if for each $X\in C$, the [[comma (infinity,1)-category]] $(X \downarrow G)$ has an [[initial object in an (infinity,1)-category|initial object]], i.e. every object $X\in C$ admits a [[universal arrow]] $X\to G F X$ to $G$.

\end{prop}
This is stated explicitly as \hyperlink{RVElements}{Riehl-Verity, Corollary 16.2.7}, and can be extracted with some work from [[Higher Topos Theory|HTT, Proposition 5.2.4.2]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

A large class of examples of $(\infty,1)$-adjunctions arises from [[Quillen adjunctions]] of [[model categories]], or adjunctions in [[sSet]]-[[enriched category theory]].

\hypertarget{quillen_adjunctions}{}\subsubsection*{{Quillen adjunctions}}\label{quillen_adjunctions}

Any [[Quillen adjunction]] induces an adjunction of [[(infinity,1)-categories]] on the [[simplicial localizations]]. See \hyperlink{Hinich14}{Hinich 14} or \hyperlink{MazelGee15}{Mazel-Gee 15}.

\hypertarget{SimplicialAndDerived}{}\subsubsection*{{Simplicial and derived adjunctions}}\label{SimplicialAndDerived}

We want to produce Cartesian/coCartesian fibration $K \to \Delta[1]$ from a given [[sSet]]-[[enriched category theory|enriched]] adjunction. For that first consider the following characterization

\begin{lemma}
\label{}\hypertarget{}{}
Let $K$ be a [[simplicially enriched category]] whose [[hom-objects]] are all [[Kan complexes]], regard the [[interval category]] $\Delta[1] := \{0 \to 1\}$ as an $sSet$-category in the obvious way using the embedding $const : Set \hookrightarrow sSet$ and consider an $sSet$-enriched functor $K \to \Delta[1]$. Let $C := K_0$ and $D := K_1$ be the $sSet$-enriched categories that are the fibers of this. Then under the [[homotopy coherent nerve]] $N : sSet Cat \to sSet$ the morphism

\begin{displaymath}
N(p) : N(K) \to \Delta[1]
\end{displaymath}
is a [[Cartesian fibration]] precisely if for all objects $d \in D$ there exists a morphism $f : c \to d$ in $K$ such that postcomposition with this morphism

\begin{displaymath}
C(c',f ) :  C(c',c) = K(c',c) \to K(c',d)
\end{displaymath}
is a [[homotopy equivalence]] of [[Kan complex]]es for all objects $c' \in C'$.

\end{lemma}
This appears as [[Higher Topos Theory|HTT, prop. 5.2.2.4]].

\begin{proof}
The statement follows from the characterization of [[Cartesian morphism]]s under homotopy coherent nerves ([[Higher Topos Theory|HTT, prop. 2.4.1.10]]), which says that for an $sSet$-enriched functor $p : C \to D$ between Kan-complex enriched categories that is [[hom-object]]-wise a [[Kan fibration]], a morphim $f : c' \to c''$ in $C$ is an $N(p)$-[[Cartesian morphism]] if for all objects $c \in C$ the diagram

\begin{displaymath}
\itexarray{
    C(c,c') &\stackrel{C(c,f)}{\to}& C(c,c'')
    \\
    \downarrow^{\mathrlap{p_{c,c'}}} &&
    \downarrow^{\mathrlap{p_{c,c''}}}
    \\
    D(p(c),p(c')) &\stackrel{D(p(c),p(f))}{\to}&
    D(p(c), p(c''))
  }
\end{displaymath}
is a [[homotopy pullback]] in the [[model structure on sSet-categories]].

For the case under consideration the functor in question is $p : K \to \Delta[1]$ and the above diagram becomes

\begin{displaymath}
\itexarray{
    K(c,c') &\stackrel{K(c,f)}{\to}& K(c,c'')
    \\
    \downarrow
     &&
    \downarrow    
    \\
    * &\to& *
  }
  \,.
\end{displaymath}
This is clearly a homotopy pullback precisely if the top morphism is an equivalence.

\end{proof}
Using this, we get the following.

\begin{prop}
\label{}\hypertarget{}{}
For $C$ and $D$ [[sSet]]-[[enriched categories]] whose hom-objects are all [[Kan complexes]], the image

\begin{displaymath}
N(C) \stackrel{\overset{N(L)}{\to}}{\underset{N(R)}{\leftarrow}}
  N(D)
\end{displaymath}
under the [[homotopy coherent nerve]] of an [[sSet]]-enriched adjunction between $sSet$-[[enriched categories]]

\begin{displaymath}
C \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
  D
\end{displaymath}
is an adjunction of [[quasi-categories]].

Moreover, if $C$ and $D$ are equipped with the structure of a [[simplicial model category]] then the quasi-categorically [[derived functors]]

\begin{displaymath}
N(C^\circ) \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
  N(D^\circ)
\end{displaymath}
form an adjunction of quasi-categories.

\end{prop}
\begin{proof}
The first part is [[Higher Topos Theory|HTT, cor. 5.2.4.5]], the second [[Higher Topos Theory|HTT, prop. 5.2.4.6]].

To get the first part, let $K$ be the $sSet$-category which is the join of $C$ and $D$: its set of objects is the disjoint union of the sets of objects of $C$ and $D$, and the [[hom-object]]s are

\begin{itemize}%
\item for $c,c' \in C$: $K(c,c') := C(c,c')$;


\item for $d,d' \in D$: $K(d,d') := D(d,d')$;


\item for $c \in C$ and $d \in D$: $K(c,d) := C(L(c),d) = D(c,R(d))$;

and

$K(d,c) = \emptyset$



\end{itemize}
and equipped with the evident composition operation.

Then for every $d \in D$ there is the morphism $Id_{R(d)} \in K(R(d),d)$, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence $N(K) \to \Delta[1]$ is a [[Cartesian fibration]].

By the analogous dual argument, we find that it is also a coCartesian fibration and hence an adjunction.

For the second statement, we need to refine the above argument just slightly to pass to the full $sSet$-subcategories on fibrant cofibrant objects:

let $K$ be as before and let $K^\circ$ be the full $sSet$-subcategory on objects that are fibrant-cofibrant (in $C$ or in $D$, respectively). Then for any fibrant cofibrant $d \in D$, we cannot just use the identity morphism $Id_{R(d)} \in K(R(d),d)$ since the right Quillen functor $R$ is only guaranteed to respect fibrations, not cofibrations, and so $R(d)$ might not be in $K^\circ$. But we can use the [[small object argument]] to obtain a functorial cofibrant replacement functor $Q : C \to C$, such that $Q(R(d))$ is cofibrant and there is an acyclic fibration $Q(R(d)) \to R(d)$. Take this to be the morphism in $K(Q(R(d)), d)$ that we pick for a given $d$. Then this does induce a homotopy equivalence

\begin{displaymath}
C(c', Q(R(d))) \to C(c',R(d)) = K(c',d)
\end{displaymath}
because in an [[enriched model category]] the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.

\end{proof}
\hypertarget{localizations}{}\subsubsection*{{Localizations}}\label{localizations}

A pair of adjoint $(\infty,1)$-functors $(L \dashv R) : C \stackrel{\leftarrow}{\hookrightarrow} D$ where $R$ is a [[full and faithful (∞,1)-functor]] exhibits $C$ as a [[reflective (∞,1)-subcategory]] of $D$. This subcategory and the composite $R \circ L : D \to D$ are a [[localization of an (∞,1)-category|localization]] of $D$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]


\item [[proadjoint]], [[Hopf adjunction]]


\item [[2-adjunction]]

[[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]


\item \textbf{adjoint $(\infty,1)$-functor}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Section 5.2 in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
A study of adjoint functors between [[quasi-categories]] is given in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{The 2-category theory of quasi-categories} (\href{http://arxiv.org/abs/1306.5144}{arXiv:1306.5144}),

\end{itemize}
and further discussion, including also that of [[(infinity,1)-monads]] is in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{Homotopy coherent adjunctions and the formal theory of monads}, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 (\href{http://arxiv.org/abs/1310.8279}{arXiv:1310.8279}, \href{https://doi.org/10.1016/j.aim.2015.09.011}{doi:10.1016/j.aim.2015.09.011})

\end{itemize}
A textbook development in the context of [[infinity-cosmoi]] can be found in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{Elements of $\infty$-category theory}, \href{http://www.math.jhu.edu/~eriehl/elements.pdf}{pdf}

\end{itemize}
The proof that a [[Quillen adjunction]] of [[model categories]] induces an adjunction of [[(∞,1)-categories]] is recorded in

\begin{itemize}%
\item [[V. Hinich]], \emph{Dwyer-Kan localization revisited}, \href{http://arxiv.org/abs/1311.4128}{arXiv:1311.4128}.

\end{itemize}
and also in

\begin{itemize}%
\item [[Aaron Mazel-Gee]], \emph{Quillen adjunctions induce adjunctions of quasicategories}, \href{http://arxiv.org/abs/1501.03146}{arXiv:1501.03146}.

\end{itemize}
[[!redirects adjoint (infinity,1)-functors]]

[[!redirects adjoint (∞,1)-functor]] [[!redirects adjoint (∞,1)-functors]]

[[!redirects adjoint ∞-functor]] [[!redirects adjoint ∞-functors]]

[[!redirects (∞,1)-adjunction]] [[!redirects (∞,1)-adjunctions]]

[[!redirects adjunction of (infinity,1)-categories]] [[!redirects adjunctions of (infinity,1)-categories]]

[[!redirects adjunction of (∞,1)-categories]] [[!redirects adjunctions of (∞,1)-categories]]

[[!redirects adjoint infinity-functor]] [[!redirects adjoint infinity-functors]]



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