$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.6.9.26. 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 cocomplete $\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 $\overline{W}$ be the collection of all $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}_0$ is a reflective full subcategory of $\operatorname{\mathcal{C}}$, the functor $L$ exhibits $\operatorname{\mathcal{C}}_0$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $\overline{W}$ (Example 6.3.3.11). It follows that, for any $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $L$ induces a fully faithful functor $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ whose essential image is spanned by the functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry every element of $\overline{W}$ to an isomorphism in $\operatorname{\mathcal{D}}$. Note that, if $\operatorname{\mathcal{D}}$ is cocomplete, then a functor $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is cocontinuous if and only if its composition with $L$ is cocontinuous (Remark 9.5.6.7). It will therefore suffice to show that if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cocontinuous functor which carries every element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$, then it carries every element of $\overline{W}$ to an isomorphism in $\operatorname{\mathcal{D}}$. This follows from Example 9.6.9.20, since $\overline{W}$ is generated by $W$ as a strongly saturated collection of morphisms (Proposition 9.6.9.23).
$\square$