Proposition 9.2.2.21. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete. The following conditions are equivalent:
The functor $F$ is $\lambda $-cocontinuous: that is, it preserves $\lambda $-small colimits.
The functor $F$ is both $\kappa $-cocontinuous and $(\kappa ,\lambda )$-finitary.
Proof. The implication $(1) \Rightarrow (2)$ is trivial (and does not require the assumption $\kappa \trianglelefteq \lambda $). For the converse, suppose that condition $(2)$ is satisfied. We wish to show that the functor $F$ preserves $K$-indexed colimits for every $\lambda $-small simplicial set $K$. Using Lemma 9.1.7.18, we can realize $K$ as the colimit of a diagram of $\kappa $-small simplicial subsets $\{ K_{\alpha } \} _{\alpha \in A}$ indexed by a $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$. If $F$ is $\kappa $-cocontinuous, then it preserves $K_{\alpha }$-indexed colimits for each index $\alpha \in A$. Our assumption that $F$ is $(\kappa ,\lambda )$-finitary guarantees that it also admits $\operatorname{N}_{\bullet }(A)$-indexed colimits. Applying Corollary 9.1.6.6, we conclude that $F$ preserves colimits indexed by the filtered colimit $K = \varinjlim _{\alpha \in A} K_{\alpha }$. $\square$