Corollary 9.1.5.13. Let $\kappa $ be an infinite cardinal and let $\operatorname{\mathcal{C}}$ be a small $\kappa $-filtered $\infty $-category. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ which carries each object $C \in \operatorname{\mathcal{C}}$ to a $\kappa $-filtered $\infty $-category $\mathscr {F}(C)$. Then the colimit $\varinjlim (\mathscr {F} )$ is also $\kappa $-filtered.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration with covariant transport representation $\mathscr {F}$. It follows from Corollary 9.1.3.11 that $\operatorname{\mathcal{E}}$ is a $\kappa $-filtered $\infty $-category. Let $W$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. By virtue of Proposition 7.4.5.1, the colimit $\varinjlim (\mathscr {F} )$ can be identified with the localization $\operatorname{\mathcal{E}}[W^{-1}]$, which is $\kappa $-filtered by virtue of Corollary 9.1.5.12. $\square$