Example 7.1.7.12. In the situation of Remark 7.1.7.11, suppose that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration. Then condition $(\ast ')$ can be reformulated as follows:
- $(\ast '')$
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the composition
\[ K^{\triangleright } \xrightarrow { \overline{q} } \operatorname{\mathcal{E}}_{C} \xrightarrow { e_{!} } \operatorname{\mathcal{E}}_{C'} \]is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C'}$. Here $e_{!}$ denotes the covariant transport functor of Notation 5.2.2.9.