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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.

Proof. The implication $(1) \Rightarrow (2)$ is trivial (and does not require the assumption $\kappa \trianglelefteq \lambda $). The reverse impllication follows from Corollary 9.1.6.6, since every $\lambda $-small $\infty $-category can be realized as a $\lambda $-small, $\kappa $-filtered colimit of $\kappa $-small $\infty $-categories (Variant 9.1.7.21). $\square$