Proposition 8.3.4.9 (Adjunctions as Profunctors). Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories which represents a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{<\kappa }$. Then a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the profunctor $\mathscr {K}$. In particular, $\mathscr {K}$ is corepresentable if and only if the functor $G$ admits a left adjoint.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose a realization of $\mathscr {K}$ as the covariant transport representation of a coupling of $\infty $-categories $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ (see Remark 8.3.2.5). By virtue of Remark 8.3.4.6, the coupling $\lambda $ is representable by the functor $G$. By virtue of Theorem 8.2.5.1, a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the coupling $\lambda $. Invoking Remark 8.3.4.6 again, we see that this is equivalent to the requirement that $F$ corepresents the profunctor $\mathscr {K}$. $\square$
Proof of Proposition 8.3.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which admit $\operatorname{Hom}$-functors
\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {H}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}. \]
Suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. It follows from Proposition 11.9.2.5 (and Remark 11.9.2.4) that $G$ is right adjoint to $F$ if and only if the profunctors $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -)$ and $\mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$ are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$). In particular, if a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresented by $F$, then it is represented by $G$ if and only if $G$ is right adjoint to $F$. $\square$