Corollary 7.1.10.7. Let $K$ be a simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cocartesian fibration of simplicial sets satisfying the following conditions:
For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ admits $K$-indexed colimits.
For every edge $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ preserves $K$-indexed colimits.
Then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ admits $K$-indexed colimits. Moreover, a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ a colimit diagram if and only if, for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $\overline{q}_{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.