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Lemma 5.3.6.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ be a scaffold. Then, for every morphism of simplicial sets $S \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map

$\lambda _{S}: S \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow S \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$

is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.7.3, we may assume without loss of generality that $S = \Delta ^ n$ is a standard simplex. Replacing $\operatorname{\mathcal{C}}$ by the category $[n] = \{ 0 < 1 < \cdots < n \}$, we are reduced to proving that $\lambda$ is a categorical equivalence, which follows from Theorem 5.3.5.7. $\square$