Prove that if $\phi : K \rightarrow L$ is a simplical approximation to $f : |K| \rightarrow |L|$, then $|\phi| \simeq f$.
I can't figure a way to show that these two functions are homotopic from just the fact that $\phi$ is a simplical approximation of $f$, or $f(\text{st}(p)) \subseteq \text{st}(\phi(p))$ where $\text{st}$ means star of point $p$.
I see how I could show it easily of $|L|$ were contractible, or $|K|$ was contractible and $|L|$ was path connected, but it isn't always the case that $|K|$ and $|L|$ satisfy these conditions.
Anyone have any ideas how to show that these two functions are homotopic?