Corollary 9.2.1.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then $G$ is fully faithful, and the essential image of $G$ is spanned by the collection of $W$-local objects of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The assertion that $G$ is fully faithful follows from Proposition 6.3.3.6. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be the unit of an adjunction between $F$ and $G$. Then an object $C \in \operatorname{\mathcal{C}}$ belongs to the essential image of $G$ if and only if the morphism $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism. This is equivalent to the requirement that, for every object $B \in \operatorname{\mathcal{C}}$, composition with $\eta _{C}$ induces a homotopy equivalence of mapping spaces $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, (G \circ F)(C) )$. We conclude by observing that $\theta $ factors as a composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B,C) \xrightarrow { F } \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(B), F(C) ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, (G \circ F)(C) ) \]
where the second map is the homotopy equivalence of Proposition 6.2.1.17. $\square$