Kerodon

Example 9.4.1.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of simplicial sets, and let $\mathbb {K}$ be a collection of simplicial sets. Then $U$ is $\mathbb {K}$-cocomplete if and only if, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$, so that $U$ is a cartesian fibration. In this case, the desired result follows from Corollary 7.1.6.22.