Corollary 4.7.9.8. Let $\kappa $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then $U$ is essentially $\kappa $-small if and only if, for every edge $e: \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{e} = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^ n$ has only finitely many edges. By virtue of Corollary 4.7.6.17, we can reduce to the case where $\kappa $ is regular. In this case, the desired result follows from Proposition 4.7.9.7 and Remark 4.7.9.6. $\square$