Example 7.6.6.36. Let $\mathbb {K}$ be a collection of weakly contractible simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an isofibration of $\infty $-categories. If $\operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete and $U$ is $\mathbb {K}$-cocontinuous, then $U$ is $\mathbb {K}$-cocomplete. To prove this, we must show that for every object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete and the inclusion functor $\iota : \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocontinuous (Example 7.6.6.35). This follows by applying Proposition 7.1.9.8 to the commutative diagram
which is a categorical pullback square by virtue of Corollary 4.5.3.28.