It seems to be well known that taking the homology of a simplicial set is homotopy invariant, meaning that a homotopy of simplicial sets induces a chain homotopy of the corresponding chain complexes, hence an isomorphism in homology.
What is the standard way to prove this? I know only May's proof [Proposition 5.3 of Simplicial objects in Algebraic Topology], which is far from elegant and uses a combinatorial, equivalent definition of homotopy between simplicial maps.
Are there any other, smoother ways? What are some references in the literature?