Proposition 6.2.1.14 (02EY)

Proposition 6.2.1.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be another functor of $\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 the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.
$(2)$: The induced map $\operatorname{id}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \rightarrow \mathrm{h} \mathit{G} \circ \mathrm{h} \mathit{F}$ is the unit of an adjunction between the homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$.

Proof. The implication $(1) \Rightarrow (2)$ follows from the observation that the formation of homotopy categories defines a (strict) functor of $2$-categories

\[ \mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}} \rightarrow \mathbf{Cat} \quad \quad \operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \]

and therefore carries adjunctions to adjunctions (see Exercise 6.1.1.6). We will show that $(2)$ implies $(1)$. By assumption, the functor $F$ admits a right adjoint $G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Let $\eta ': \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow F \circ G'$ be the unit of an adjunction. Applying Corollary 6.1.3.3, we deduce that there exists a natural transformation $\gamma : G' \rightarrow G$ such that $\eta $ is a composition of the natural transformations

\[ \eta ': \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow F \circ G' \quad \quad (\operatorname{id}_ F \circ \gamma ): F \circ G' \rightarrow F \circ G \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. If assumption $(2)$ is satisfied, then the image of $\gamma $ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{D}}}, \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an isomorphism: that is, $\gamma $ carries each object $D \in \operatorname{\mathcal{D}}$ to an isomorphism $\gamma _{D}: G'(D) \rightarrow G(D)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Applying Theorem 4.4.4.4, we conclude that $\gamma $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$, so that the criterion of Corollary 6.1.3.3 guarantees that $\eta $ is also the unit of an adjunction. $\square$