I'm reading this proof of the statement above, and get stuck on the part about the surjectivity of the homomorphism $\phi$. My understanding of the argument is that, given a loop $\alpha$ in $|K|$, the simplicial approximation theorem (attached below) should produce an approximation $s$ to $\alpha$, which is the representative of an element of $E(K,v)$, hence that should prove the theorem. However, the approximation theorem doesn't say much about what the map $s$ would be like, whereas $s$ must have some very specific properties to make the argument work. I'd like to have some elaboration on the argument.
Another question in the part about the injectivity of $\phi$: why does the homotopy send the three sides of the square to $v$?
Attached are the proof in question, and some other relevant definitions and theorems. Note that $|L^m|$ is the notation for the polyhedron of the simplicial complex $L$, after being barycentrically subdivided $m$ times. Also the polyhedron of a simplicial complex is just that complex, seen as a topological space.
