Definition of a geometric realization of a simplicial space.

Question

I am a little confused with the definition of a geometric realization. I follow "geometric realization of a simplicial topological spaces" link on ncatlab. It says that $|X|=\coprod_n X_n\times \Delta^n$ quotient out by the following relation: $(x,f_*(p))\sim (f^*(x),p)$ for each $f:[k]\rightarrow [l]$. I guess $f_*:\Delta^k \rightarrow \Delta^{k+1}$ is the inclusion of a k-face. Then what is the $f_*:\Delta^{k+1} \rightarrow \Delta^{k}$ ? Is it simply the linear projection to the k-face so that a chosen vertex maps to the center of a k-face?