Kerodon

Corollary 7.1.7.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cocartesian fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ be a morphism of simplicial sets. Then $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ which is preserved by the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$, for every morphism $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$.