Kerodon

Example 9.2.2.16. Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume either that $\operatorname{\mathcal{C}}$ is idempotent-complete or that $\kappa = \aleph _0$, so that $\operatorname{\mathcal{C}}$ is $(\kappa ,\kappa )$-cocomplete (Proposition 9.2.1.14). Then $F$ is automatically $(\kappa ,\kappa )$-finitary: that is, it preserves $\operatorname{\mathcal{K}}$-indexed colimits for every $\infty $-category $\operatorname{\mathcal{K}}$ which is both $\kappa $-small and $\kappa $-filtered. For $\kappa = \aleph _0$, this follows from Corollary 9.1.8.12. If $\kappa $ is uncountable, we can use Proposition 9.1.8.11 to reduce to the case $\operatorname{\mathcal{K}}= \operatorname{N}_{\bullet }(\operatorname{Idem})$, in which case the desired result is automatic (Corollary 8.5.3.12).