Constructing a simplicial map from a diagram

This simply contracts $\Delta^n$ onto the $0$-vertex. The diagram you draw encodes the following map $\mathbf{n} \times\mathbf{1} \to \mathbf{n}$ where elements $(m,0)\mapsto 0$ and $(m,1) \mapsto m$. This map induces $$ \Delta^n\times \Delta^1 = \operatorname{hom}_\Delta(-, \mathbf{n}) \times \operatorname{hom}_\Delta(-, \mathbf{1}) \cong \operatorname{hom}_\Delta(-,\mathbf{n} \times\mathbf{1}) \to \operatorname{hom}_\Delta(-, \mathbf{n}) = \Delta^n. $$ It could help if you check the effect after geometric realization. This is the standard contracting homotopy of $|\Delta^n|$ to $|\Delta^0| = *$. Write a formula for this homotopy and compare!

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.