Kerodon

Proposition 11.9.2.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between locally small $\infty $-categories and let $\eta : \operatorname{id}_{ \operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. Then $\eta $ exhibits $G$ as a right adjoint to $F$ if and only if the morphism $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -) \rightarrow \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$ of Remark 11.9.2.4 is an isomorphism of profunctors.