Remark 8.6.5.10. In the situation of Proposition 8.6.5.9, let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$ and let
\[ e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'} \quad \quad e'_{!}: \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}}) \]
be functors given by covariant transport along $e$ (for the cocartesian fibrations $U$ and $\pi $, respectively). Then the functor $e'_{!}$ is given by left Kan extension along $e_{!}$.