Let $\Delta$ be the simplex category with objects $[n]=\{0,...,n\}$, $n\geq 0$, and morphisms ordering preserving functions $[n]\rightarrow [m]$. Then let the standard categorical $n$-simplex be $Hom_{\Delta}(\bullet,[n])$, and is a simplicial set.
It is well known that the geometric realization of a categorical $n$-simplex is the standard topological $n$-simplex, but I can't seem to find a proof of this fact.
Question: Why is the geometric realization of a categorical $n$-simplex the standard topological $n$-simplex?