Corollary 9.1.5.12. Let $\kappa $ be an infinite cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category. Then, for any collection of morphisms $W$ of $\operatorname{\mathcal{C}}$, the localization $\operatorname{\mathcal{C}}[W^{-1}]$ is $\kappa $-filtered.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ be a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Definition 6.3.1.9). Since $F$ is right cofinal (Proposition 7.2.1.10), Corollary 9.1.5.11 guarantees that $\operatorname{\mathcal{C}}[W^{-1}]$ is a $\kappa $-filtered $\infty $-category. $\square$