Kerodon

Remark 7.1.7.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is both locally cartesian and locally cocartesian, and fix a vertex $C \in \operatorname{\mathcal{C}}$. Example 7.1.7.12 asserts that every colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is an edgewise $U$-colimit diagram. Combining this observation with Example 7.1.7.12, we see that for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ preserves $K$-indexed colimits. This is a special case of Corollary 7.1.4.28, since the functor $e_{!}$ admits a right adjoint (given by contravariant transport along $e$: see Proposition 6.2.5.4).