Corollary 6.2.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\eta $ is the unit of an adjunction between $F$ and $G$.
- $(2)$
For every pair of objects $X \in \operatorname{\mathcal{C}}$ and $Y \in \operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (G \circ F)(X), G(Y) ) \xrightarrow { \circ [ \eta _ X]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ) \]is a homotopy equivalence of Kan complexes.
- $(3)$
The functor $F$ admits a right adjoint. Moreover, for every pair of objects $X \in \operatorname{\mathcal{C}}$ and $Y \in \operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{D}}}}( F(X), Y) \xrightarrow {G} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( (G \circ F)(X), G(Y) ) \xrightarrow { \circ [ \eta _ X]} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, G(Y) ) \]is a bijection of sets.