Proposition 9.1.6.1. Let $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\kappa $ be an infinite cardinal. If $\operatorname{\mathcal{D}}$ is $\kappa $-filtered and $F$ is right cofinal, then $\operatorname{\mathcal{C}}$ is also $\kappa $-filtered.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\lambda $-small. It follows from Proposition 9.1.5.8 (together with Proposition 9.1.4.1) that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered if and only if, for every $\kappa $-small simplicial set $L$, the limit functor $\varprojlim : \operatorname{Fun}(L, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ preserves $\operatorname{\mathcal{C}}$-indexed colimits. Since $F$ is right cofinal, it will suffice to show that $\varprojlim $ preserves $\operatorname{\mathcal{D}}$-indexed colimits, which follows from our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-filtered. $\square$