Proposition 9.1.9.17. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:
If $\kappa $ is uncountable, then $\operatorname{\mathcal{C}}$ admits $\kappa $-small $\kappa $-filtered colimits if and only if it is idempotent complete.
If $\kappa = \aleph _0$, then $\operatorname{\mathcal{C}}$ automatically admits $\kappa $-small $\kappa $-filtered colimits.
In either case, any functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa , \kappa )$-finitary.