Example 5.2.2.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between ordinary categories, let $f: C \rightarrow D$ be a morphism in $\operatorname{\mathcal{C}}$, and choose a collection of $U$-cocartesian morphisms $\{ \widetilde{f}_{X}: X \rightarrow f_{!}(X) \} _{X \in \operatorname{\mathcal{E}}_{C} }$ satisfying $U( \widetilde{f}_{X} ) = f$. According to Proposition 5.2.2.1, there is a unique functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ for which the construction $X \mapsto \widetilde{f}_{X}$ determines a natural transformation of functors $\widetilde{f}: \operatorname{id}_{ \operatorname{\mathcal{E}}_{C} } \rightarrow f_{!}$. Passing to nerves, we obtain a natural transformation $\operatorname{id}_{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}_{C} )} \rightarrow \operatorname{N}_{\bullet }(f_!)$, which exhibits the functor
as given by covariant transport along $f$ (regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$).
Stated more informally, the covariant transport construction for cocartesian fibrations of ordinary categories (see Construction 5.2.2.2) can be regarded as a special case Definition 5.2.2.4.