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Proposition 5.2.6.19. 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 $U$ is a categorical equivalence of simplicial sets.

Proof of Proposition 5.2.6.19. We proceed by induction on $n$. If $n=0$, then $U$ is an isomorphism and there is nothing to prove. Assume that $n \geq 1$, let $\operatorname{\mathcal{C}}(\geq 1)$ denote the fiber product $\operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times _{\Delta ^ n} \operatorname{\mathcal{C}}$, and let $A \subseteq \Delta ^ n$ be the simplicial subset given by the pushout

\[ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \coprod _{ \{ 1\} } \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ). \]

Since the inclusion map $\{ 1\} \hookrightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} )$ is left anodyne (Lemma 4.3.7.8), it follows from Example 4.3.6.5 that the inclusion map $A \hookrightarrow \Delta ^ n$ is inner anodyne. Applying Lemma 5.2.6.17, we deduce that the inclusion

\[ A \times _{\Delta ^ n} M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n)) \hookrightarrow M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \]

is also inner anodyne. It will therefore suffice to show that the restriction of $U$ to $A \times _{\Delta ^ n} M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n))$ is a categorical equivalence. Equivalently, we wish to show that the outer rectangle of the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}(0) \ar [r] \ar@ {=}[d] & M( \operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{C}}(2) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \ar [d] \\ \{ 1\} \times \operatorname{\mathcal{C}}(0) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(\geq 1) \ar [d] \\ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{\mathcal{C}}(0) \ar [r]^-{h} & \operatorname{\mathcal{C}}} \]

is a categorical pushout square. Our inductive hypothesis guarantees that the upper vertical maps in this diagram are categorical equivalences, so that the upper square is a categorical pushout (Proposition 4.5.3.7). By virtue of Proposition 4.5.3.5, it will suffice to show that the lower square is a categorical pushout diagram. Let $\rho : \Delta ^{n} \rightarrow \Delta ^1$ be the morphism given on vertices by the formula $\rho (i) = \begin{cases} 0 & \textnormal{ if } i = 0 \\ 1 & \textnormal{ if } i > 0.\end{cases}$ Then $\rho $ is a cocartesian fibration of simplicial sets, and the edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Delta ^ n$ is $\rho $-cocartesian. Note that $\operatorname{\mathcal{C}}(\geq 1)$ can be identified with the inverse image $(\rho \circ \pi )^{-1} \{ 1\} $. For every object $C \in \operatorname{\mathcal{C}}(0)$, it follows from Remark 5.1.1.6 that the composition

\[ \Delta ^1 \times \{ C\} \hookrightarrow \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{\mathcal{C}}(0) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $(\rho \circ \pi )$-cocartesian edge of $\operatorname{\mathcal{C}}$. The desired result now follows by applying the criterion of Theorem 5.2.5.1. $\square$