Proposition 8.7.1.16. Let $\kappa $ be an uncountable regular cardinal and suppose we are given a commutative diagram of $\infty $-categories
where $U$ and $\widehat{U}$ are inner fibrations and $U$ is essentially $\kappa $-small. Then (8.86) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ (in the sense of Definition 8.7.1.12) if and only if the following conditions are satisfied:
- $(1)$
For each object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{E}}_{C}$.
- $(2)$
The functor $H$ is fully faithful.
- $(3)$
The inner fibration $\widehat{U}$ is locally cartesian.
- $(4)$
For each morphism $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }: \widehat{\operatorname{\mathcal{E}}}_{C'} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ preserves $\kappa $-small colimits.