Lemma 9.5.4.19. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an edgewise accessible inner fibration (Definition 9.4.8.16). Assume that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is presentable. If $U$ is either a locally cartesian fibration or locally cocartesian fibration, then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is presentable.
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Proof. It follows from Corollary 9.4.8.27 that the fibration $U$ is edgewise accessible, so that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is accessible (Corollary 9.4.8.33). Applying Corollary 7.1.10.5, we see that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is either complete (if $U$ is a locally cocartesian fibration) or cocomplete (if $U$ is a locally cartesian fibration), and is therefore presentable. $\square$