Remark 4.7.9.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa $ be an uncountable regular cardinal. Then $U$ is essentially $\kappa $-small if and only if it is locally $\kappa $-small and, for each vertex $C \in \operatorname{\mathcal{C}}$, the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } )$ is $\kappa $-small. See Proposition 4.7.8.7.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$