Remark 11.9.8.4 (0259): Compatibility with Pullback

Remark 11.9.8.4 (Compatibility with Pullback). Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a functor 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 $. For every morphism of simplices $\alpha : \Delta ^{m} \rightarrow \Delta ^{n}$, the pullback

\[ \Delta ^{m} \times _{ \Delta ^{n} } M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {\operatorname{id}\times U} \Delta ^{m} \times _{\Delta ^ n} \operatorname{\mathcal{C}} \]

is a scaffold of the projection map $\Delta ^{m} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$; here we implicitly invoke Remark 11.6.0.160 to identify $\Delta ^{m} \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) )$ with the mapping simplex of the diagram

\[ \operatorname{\mathcal{C}}( \alpha (0) ) \rightarrow \operatorname{\mathcal{C}}( \alpha (1) ) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}( \alpha (m) ). \]