Corollary 9.4.8.13. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a small collection of accessibly embedded subcategories of $\operatorname{\mathcal{C}}$. Then the intersection $\bigcap _{i \in I} \operatorname{\mathcal{C}}_ i$ is an accessibly embedded subcategory of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Example 9.4.7.4, we see that the product $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is an accessibly embedded subcategory of the product $\prod _{i \in I} \operatorname{\mathcal{C}}$. The desired result now follows from Corollary 9.4.8.12, since the intersection $\bigcap _{i \in I} \operatorname{\mathcal{C}}_ i$ is the inverse image of $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ along the diagonal functor $\operatorname{\mathcal{C}}\rightarrow \prod _{i \in I} \operatorname{\mathcal{C}}$. $\square$