Corollary 11.9.6.6. 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.