Corollary 5.6.7.7. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. The following conditions are equivalent:
- $(1)$
The simplicial set $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.
- $(2)$
For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.