TITLE: Constructing a simplicial map from a diagram
QUESTION [5 upvotes]: I am trying to read Goerss and Jardine's book (Simplicial Homotopy Theory) and in the proof of Theorem 7.10 (in chapter 1), they claim that there is a simplicial homotopy $\Delta^n\times\Delta^1\to\Delta^n$ given by the diagram:
$$\begin{array}{ccccccc}
0&\to &0&\to &\dots&\to &0\\
\downarrow &&\downarrow &&\dots &&\downarrow\\
0&\to &1&\to&\dots&\to &n
\end{array}$$
I am unsure how this actually defines a simplicial map at all. Is there a way to construct a simplicial map from this diagram?
If it helps, the simplicial map is (I think) supposed to be a contraction from the identity to 0.

REPLY [2 votes]: Remember that a map between products of simplices is nothing more than a map of the corresponding partially ordered sets. Thus what is given here is the values of the desired map of the vertices of $\Delta^n\times \Delta^1$, which uniquely determine the map.