Remark 9.2.9.13. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{E}}$ be an $\infty $-category equipped with a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$. The following conditions are equivalent:
- $(1)$
The inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is $(\kappa ,\lambda )$-cocomplete, in the sense of Definition 9.2.9.4.
- $(2)$
For $0 \leq i \leq n$, the fiber $\operatorname{\mathcal{E}}_ i = U^{-1} \{ i\} $ is $(\kappa ,\lambda )$-cocomplete, and the inclusion functor $\operatorname{\mathcal{E}}_ i \hookrightarrow \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-finitary.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete and each of the full subcategories $\operatorname{\mathcal{E}}_{i} \subseteq \operatorname{\mathcal{E}}$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits.
If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ is $(\kappa , \lambda )$-finitary if and only if the restriction $F|_{ \operatorname{\mathcal{E}}_{i} }: \operatorname{\mathcal{E}}_ i \rightarrow \operatorname{\mathcal{E}}'$ is $(\kappa ,\lambda )$-finitary for $0 \leq i \leq n$.