$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.5.7.2. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a left adjoint to the inclusion functor. For every presentable $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $L$ determines a fully faithful functor
\[ L^{\ast }: \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}), \]
whose essential image is spanned by the cocontinuous functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$.
Proof.
Let $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ spanned by the continuous accessible functors, and define $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}_0 )$ similarly. Using Theorem 9.5.2.1 and Corollary 8.3.4.10, we can identify $L^{\ast }$ with the opposite of the inclusion functor $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}_0 ) \hookrightarrow \operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$. It follows immediately that $L^{\ast }$ is fully faithful. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocontinuous functor, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a right adjoint of $F$. Then $F$ belongs to the essential image of $L^{\ast }$ if and only if the functor $G$ carries each object $X \in \operatorname{\mathcal{D}}$ to a $W$-local object of $\operatorname{\mathcal{C}}$: that is, for every morphism $w: C \rightarrow D$ of $W$, the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, G(X) ) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(X) )$ is a homotopy equivalence of Kan complexes. This is equivalent to the condition that composition with $[F(w)]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(D), X ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), X)$, which is satisfied for every $X \in \operatorname{\mathcal{D}}$ if and only if $F(w)$ is an isomorphism.
$\square$