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Corollary 5.2.6.21. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$, and let

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}. \]

be a scaffold of $\pi $. Then, for every morphism of simplicial sets $X \rightarrow \Delta ^ n$, the induced map

\[ U': X \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow X \times _{\Delta ^ n} \operatorname{\mathcal{C}} \]

is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.4.4, we may assume without loss of generality that $X = \Delta ^ m$ is a standard simplex. In this case, we can identify $U'$ with a scaffold of the projection map $\Delta ^ m \times _{\Delta ^ n} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$ (Remark 5.2.6.16), so the desired result follows from Proposition 5.2.6.19. $\square$