Corollary 9.6.3.12. 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
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 $\operatorname{\mathcal{D}}$.