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Lemma 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, 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 $\square$