Proposition 9.1.9.22. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete and admits $\lambda $-small $\kappa $-filtered colimits.
Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\lambda $-small colimits if and only if it is $(\kappa ,\lambda )$-finitary and preserves $\kappa $-small colimits.