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

Remark (Functoriality in $\Delta ^ n$). Let $\alpha : [m] \rightarrow [n]$ be a nondecreasing function of linearly ordered sets. Let $\overrightarrow {X}$ be a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

and let $\overrightarrow {X}'$ denote the diagram

\[ X( \alpha (0) ) \rightarrow X( \alpha (1) ) \rightarrow \cdots \rightarrow X( \alpha (m) ) \]

obtained by precomposition with $\alpha $. Then the mapping simplices of $\overrightarrow {X}$ and $\overrightarrow {X}'$ fit into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ M( \overrightarrow {X}' ) \ar [r] \ar [d] & M( \overrightarrow {X} ) \ar [d] \\ \Delta ^{m} \ar [r]^-{ \alpha } & \Delta ^ n. } \]