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Exercise Let $\operatorname{\mathcal{C}}$ be a category, let $n$ be an integer, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. Assume that, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $(n,1)$-category (Definition Show that the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is $n$-categorical, in the sense of Definition

In particular, if each of the simplicial sets $\mathscr {F}(C)$ is (isomorphic to) the nerve of an ordinary category, then the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is also isomorphic to the nerve of an ordinary category. For a more precise statement, see Example