Definition 7.1.7.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $C \in \operatorname{\mathcal{C}}$ be a vertex. We say that a morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an edgewise $U$-colimit diagram if it satisfies the following condition:
- $(\ast )$
For every edge $e: C \rightarrow C'$ of the simplicial set $\operatorname{\mathcal{C}}$, the composite map
\[ K^{\triangleright } \xrightarrow { \overline{f} } \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]is a colimit diagram in the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.