Remark 9.2.9.8. Let $\kappa \leq \lambda $ be regular cardinals and let $\mathbb {K}$ be the collection of all $\lambda $-small $\kappa $-filtered $\infty $-categories. Proposition 9.2.9.7 asserts that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete (in the sense of Definition 9.2.9.4) if and only if it is $\mathbb {K}$-cocomplete (in the sense of Definition 7.6.6.20). In particular:
If $U$ is $(\kappa ,\lambda )$-cocomplete, then each fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a $(\kappa ,\lambda )$-cocomplete $\infty $-category (Example 7.6.6.31). The converse holds if $U$ is a locally cartesian fibration, but not in general (Example 7.6.6.30).
If $U$ is a locally cocartesian fibration, then it is $(\kappa ,\lambda )$-cocomplete if and only if each fiber $\operatorname{\mathcal{E}}_{C}$ is a $(\kappa ,\lambda )$-cocomplete $\infty $-category and the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is $(\kappa ,\lambda )$-finitary, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ (see Example 7.6.6.32).
If $U$ is an essentially $\mu $-small cocartesian fibration, then it is $(\kappa ,\lambda )$-cocomplete if and only if the covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ factors through the subcategory $\operatorname{\mathcal{QC}}^{(\kappa ,\lambda )-\mathrm{ccomp}}_{< \mu }$ of Notation 9.2.2.14 (Example 7.6.6.33).