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Proposition 5.5.3.18. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Assume that, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. Then the comparison map

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

of Construction 5.5.3.16 is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.4.4, it will suffice to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map

\[ F_{\sigma }: \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }( \operatorname{Set_{\Delta }}) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

is a categorical equivalence of simplicial sets. Let us identify $\sigma $ with a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, so that the fiber product $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }( \operatorname{Set_{\Delta }})$ is the mapping simplex $M( \mathscr {F}(C_0) \rightarrow \cdots \rightarrow \mathscr {F}(C_ n) )$ introduced in Construction 5.2.6.3. Let us abuse notation by identifying each $\mathscr {F}(C_ i)$ with the fiber of the projection map

\[ \pi : \Delta ^ n \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \Delta ^ n \]

over the $i$th vertex of $\Delta ^ n$. It follows from Proposition 5.5.3.13 that $\pi $ is a cocartesian fibration and that $F_{\sigma }$ is a scaffold of $\pi $, in the sense of Definition 5.2.6.12. Applying Proposition 5.2.6.19, we conclude that $F_{\sigma }$ is a categorical equivalence of simplicial sets. $\square$