$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:


The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.


For every object $D \in \operatorname{\mathcal{D}}$, the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \]

is representable by an object of $\operatorname{\mathcal{C}}$.

Proof. We first show that $(1)$ implies $(2)$. Suppose that there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which exhibits $G$ as a right adjoint of $F$. Let $D$ be an object of $\operatorname{\mathcal{D}}$, so that $\epsilon $ determines a morphism $\epsilon _ D: (F \circ G)(D) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Proposition guarantees that, for every object $C \in \operatorname{\mathcal{C}}$, the natural map

\begin{eqnarray*} \{ \epsilon _ D \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) & \hookrightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) )\\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D) \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), (F \circ G)(D) ) \\ & \xrightarrow {\circ } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \end{eqnarray*}

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $G(D)$ represents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$.

We now show that $(2)$ implies $(1)$. Choose a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ having fibers $\operatorname{\mathcal{C}}\simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$, $\operatorname{\mathcal{D}}\simeq \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$, and for which the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is given by covariant transport along the nondegenerate edge of $\Delta ^1$ (for example, we can take $\operatorname{\mathcal{E}}$ to be the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$: see Proposition Assume that condition $(2)$ is satisfied. We will show that $U$ is a cartesian fibration, so that $F$ admits a right adjoint by virtue of Proposition Fix an object $D \in \operatorname{\mathcal{D}}$; we wish to show that there exists an object $X \in \operatorname{\mathcal{C}}$ and a $U$-cartesian morphism $f: X \rightarrow D$ in $\operatorname{\mathcal{E}}$. For each $C \in \operatorname{\mathcal{C}}$, we have a canonical isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C,D)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Consequently, condition $(2)$ guarantees that the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C,D) \]

is representable. In other words, there exists an object $X \in \operatorname{\mathcal{C}}$ and a morphism $f: X \rightarrow D$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, composition with the homotopy class $[f]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C, D)$. Applying Corollary, we conclude that $f: X \rightarrow D$ is $U$-cartesian. $\square$