Variant 4.7.9.11. Let $\lambda $ be an uncountable cardinal, let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. If the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small and $U$ is essentially $\lambda $-small, then the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is essentially $\lambda $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. As in the proof of Proposition 5.3.8.11, we may assume that $U$ is minimal, which guarantees that $\operatorname{\mathcal{E}}$ is $\lambda $-small. Applying Corollary 4.7.4.19, we conclude that $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-small. In particular, the simplicial subset $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-small, and therefore essentially $\lambda $-small. $\square$