$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $f: X \rightarrow S$ be a morphism of simplicial sets. Let $s$ be a vertex of $S$, which we will identify (via Proposition with its image in $\operatorname{Ex}^{\infty }(S)$. Then the canonical map $\operatorname{Ex}^{\infty }( X_ s ) \rightarrow \operatorname{Ex}^{\infty }(X)_{s}$ is an isomorphism of simplicial sets. Here $X_ s = \{ s\} \times _{S} X$ denotes the fiber of $f$ over the vertex $s$, and $\operatorname{Ex}^{\infty }(X)_{s} = \{ s\} \times _{ \operatorname{Ex}^{\infty }(S) } \operatorname{Ex}^{\infty }(X)$ is defined similarly.