Kerodon

Example 7.3.9.3. In the situation of Proposition 7.3.9.2, suppose that that $U$ is also a cartesian fibration. Then, for every morphism $e: D \rightarrow D'$ of $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}$ has a right adjoint $e^{\ast }$, given by contravariant transport along $e$ (Proposition 6.2.3.4). In particular, the functor $e_{!}$ automatically preserves $K$-indexed colimits (Corollary 7.1.4.22). We therefore recover the criterion of Corollary 7.1.6.22: the morphism $f$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$.