Corollary 7.1.10.4. Let $K$ be a simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets satisfying the following condition:
- $(\ast )$
For each 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. Moreover, every colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is an edgewise $U$-colimit diagram in $\operatorname{\mathcal{E}}$.
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 $\operatorname{\mathcal{E}}_{C}$.