Proposition 9.5.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a presentable $\infty $-category. The following conditions are equivalent:
- $(1)$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a presentable fibration (Definition 9.5.3.5). That is, it is a locally cocartesian fibration having the property that, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is cocontinuous.
- $(2)$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is accessible and continuous.