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Corollary Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Then:


The projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets.


Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark Then $(f,e)$ is $U$-cocartesian if and only if $e: \mathscr {F}(f)(x) \rightarrow y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$.

In particular, $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is an $\infty $-category.

Proof. Apply Proposition in the special case where $\mathscr {G}$ is the constant diagram taking the value $\Delta ^0$. $\square$