Definition 7.4.1.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of simplicial sets, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a diagram, and let $\alpha : \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}}$ be a natural transformation. For each vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $\alpha $ to the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ determines a natural transformation from the constant functor taking the value $\Delta ^0$ to the constant functor taking the value $\mathscr {F}(C)$, which we can identify with a morphism of Kan complexes
We say that $\alpha $ exhibits $\mathscr {F}$ as a covariant transport representation for $U$ if, for every vertex $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}$ is a homotopy equivalence.