Corollary 5.3.8.8 (05PV)

Corollary 5.3.8.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by a category $\operatorname{\mathcal{C}}$. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a minimal $\infty $-category. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a minimal inner fibration.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= [n]$ is a linearly ordered set. In this case, the desired result is a reformulation of Proposition 5.3.8.7 (see Example 5.3.8.3). $\square$