Kerodon

Example 7.6.6.30. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of simplicial sets. Then $U$ is a $\mathbb {K}$-cocomplete (in the sense of Definition 7.6.6.28) 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 (in the sense of Definition 7.6.6.20). See Example 7.1.7.10.