Kerodon

Variant 9.4.1.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\mathbb {K}$ be a collection of simplicial sets. We say that $U$ is $\mathbb {K}$-cocomplete if, for every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is a $\mathbb {K}$-cocomplete inner fibration of $\infty $-categories, in the sense of Definition 9.4.1.1. By virtue of Proposition 9.4.1.3, this agrees with Definition 9.4.1.1 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.