Definition 9.5.3.5 (Presentable Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is a presentable fibration if it satisfies the following conditions:
- $(a)$
The morphism $U$ is a locally cocartesian fibration of simplicial sets.
- $(b)$
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.
- $(c)$
For every 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.