Corollary 9.5.7.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Bousfield localization functor. Then $F$ is an epimorphism in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ (see Variant 4.7.5.3).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Remark 9.5.6.11 and Proposition 9.5.7.5, we may assume without loss of generality that there exists a small collection $W$ of morphisms of $\operatorname{\mathcal{C}}$ such that $\operatorname{\mathcal{D}}$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects and $F$ is left adjoint to the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$. In this case, the result follows from the universal property of Proposition 9.5.7.2. $\square$