Example 7.4.1.11. Let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ be the nerve of a category $\operatorname{\mathcal{C}}_0$, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be the (homotopy coherent) nerve of a functor $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$, and let $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0)$ denote the weighted nerve of Definition 5.3.3.1, so that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration of $\infty $-categories (Corollary 5.3.3.19). Restricting the natural transformation $\alpha $ of Example 7.4.1.10 along the comparison map
of Construction 5.6.4.1, we obtain a natural transformation $\alpha ': \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}}$ which exhibits $\mathscr {F}$ as a covariant transport representation for $U$ (see Proposition 5.6.4.8). Beware that, although the functors $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ and $\mathscr {F}|_{\operatorname{\mathcal{E}}}$ can be obtained from strictly commutative diagrams $\mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \operatorname{Kan}$, the natural transformation $\alpha $ generally cannot be obtained from a natural transformation of $\operatorname{Kan}$-valued functors (even up to homotopy).