Corollary 9.5.5.12 (06NX)

Corollary 9.5.5.12. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $\kappa $-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\lambda $-cocontinuous if and only if it satisfies the following pair of conditions:

$(a)$: The functor $F$ is $(\kappa ,\lambda )$-finitary.
$(b^{-})$: The restriction $F|_{ \operatorname{\mathcal{C}}_{< \kappa } }$ is $\kappa $-cocontinuous, where $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denotes the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$.

Proof. The necessity of conditions $(a)$ and $(b^{-})$ is clear. The sufficiency follows from Corollary 9.5.5.10, since $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_{< \kappa }$ (Proposition 9.4.1.11). $\square$