Corollary 4.7.9.11. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then $U$ is locally $\kappa $-small if and only if, for every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The “only if” direction is immediate from the definitions. To prove the converse, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, in which case the desired result follows from Proposition 4.7.9.5. $\square$