Remark 9.5.4.20. In the situation of Lemma 9.5.4.19, if $U$ is a locally cartesian fibration then small colimits in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ are computed levelwise: that is, they are preserved by the evaluation functor
\[ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \{ C\} , \operatorname{\mathcal{E}}) = \operatorname{\mathcal{E}}_{C} \]
for each vertex $C \in \operatorname{\mathcal{C}}$. See Corollary 7.1.10.5. Similarly, if $U$ is a locally cocartesian fibration, then small limits in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ are computed levelwise.