Corollary 9.5.6.19. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a continuous accessible functor. If $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$, then the inverse image $\operatorname{\mathcal{D}}_0 = G^{-1} \operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Corollary 9.4.8.12 that $\operatorname{\mathcal{C}}_0$ is an accessibly embedded subcategory of $\operatorname{\mathcal{C}}$. By virtue of Proposition 9.5.6.15, it will suffice to show that $\operatorname{\mathcal{C}}_0$ is closed under small limits in $\operatorname{\mathcal{C}}$. This follows from our assumption that $G$ is continuous, since $\operatorname{\mathcal{D}}_0$ is closed under small limits in $\operatorname{\mathcal{D}}$. $\square$