Example 5.6.5.6 (Weighted Nerves). Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor, and let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the weighted nerve of Definition 5.3.3.1. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (Corollary 5.3.3.16). Moreover, the equivalence
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]
of Proposition 5.6.4.8 exhibits $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ as a covariant transport representation for $U$.