Definition 9.4.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small inner fibration of $\infty $-categories. We say that a commutative diagram of $\infty $-categories
exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$ if the following conditions are satisfied:
- $(1)$
For every object $C \in \operatorname{\mathcal{C}}$, the map of fibers $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a cocompletion of $\operatorname{\mathcal{E}}_{C}$ (see Definition 8.4.0.1).
- $(2)$
The functor $H$ is fully faithful.
- $(3)$
The functor $\widehat{U}$ is a locally cartesian fibration.
- $(4)$
For every morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }: \widehat{\operatorname{\mathcal{E}}}_{D} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ preserves small colimits.