Corollary 9.5.6.21. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\{ \operatorname{\mathcal{C}}_ s \subseteq \operatorname{\mathcal{C}}\} _{s \in S}$ be a collection of Bousfield localizations of $\operatorname{\mathcal{C}}$ indexed by a small set $S$. Then the intersection $\bigcap _{s \in S} \operatorname{\mathcal{C}}_{s}$ is also a Bousfield localization of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The intersection $\bigcap _{s \in S} \operatorname{\mathcal{C}}_ s$ can be identified with the inverse image of the product $\prod _{s \in S} \operatorname{\mathcal{C}}_ s$ under the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \prod _{s \in S} \operatorname{\mathcal{C}}$. By virtue of Corollary 9.5.6.19, it will suffice to show that $\prod _{s \in S} \operatorname{\mathcal{C}}_ s$ is a Bousfield localization of $\prod _{s \in S} \operatorname{\mathcal{C}}$, which is a special case of Remark 9.5.6.13. $\square$