Corollary 7.1.10.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of simplicial sets and suppose we are given a diagram $q: K \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $q_ C: K \rightarrow \operatorname{\mathcal{E}}_{C}$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{E}}_ C$. Then:
The diagram $q$ admits a colimit in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
An extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of $q$ is 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 $\operatorname{\mathcal{E}}_{C}$.