Corollary 9.5.5.10. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, so that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\lambda $-cocomplete (Corollary 9.5.5.6). Suppose we are given a functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-cocomplete, and let $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ be the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$. The following conditions are equivalent:
- $(1)$
The functor $f$ is $\kappa $-cocontinuous.
- $(2)$
The functor $F$ is $\kappa $-cocontinuous.
- $(3)$
The functor $F$ is $\lambda $-cocontinuous.