Remark 7.4.4.10. In the situation of Construction 7.4.4.9, let $C \in \operatorname{\mathcal{C}}$ be a vertex and let $\operatorname{ev}_{C}: \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ be the evaluation functor, given on objects by $\operatorname{ev}_{C}(F) = F(C)$. Then the composition
\[ \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \xrightarrow {\mathrm{Df}} \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow {\operatorname{ev}_ C} \operatorname{\mathcal{E}}_{C} \]
is given by covariant transport along the unique edge ${\bf 0} \rightarrow C$ of $\operatorname{\mathcal{C}}^{\triangleleft }$.