Corollary 9.4.8.33. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is edgewise accessible. If $\operatorname{\mathcal{C}}$ is small, then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is accessible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Remark 9.4.8.20, we can choose a small regular cardinal $\kappa $ such that $U$ is edgewise $\kappa $-accessible. Let $\lambda $ be another small regular cardinal satisfying $\kappa \triangleleft \lambda $. Then the inner fibration $U$ is also edgewise $\lambda $-accessible (Proposition 9.4.6.5). Applying Proposition 9.4.8.32, we conclude that $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-accessible. $\square$