Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all finite intersections $U_{i_1} \cap U_{i_2} \cap \dots \cap U_{i_k}$ are either empty or contractible.

This induces a simplicial topological space $U_{\bullet}$. Let $Z_{\bullet}$ be another simplicial topological space. I keep reading that any continuous map $$ f \colon |U_{\bullet}| \to |Z_{\bullet}| $$ is homotopic to a simplicial one (e.g. in the sketched proof of theorem 4.5 here). Is there any reference for this? How do I prove this?