On the construction of a covering map

Let {Y=\lvert X\rvert} be the geometric realization of a path-connected simplicial set X, and let {G=\pi_{1}(X)} be the fundamental group. Given a subgroup {H\subset G} , let {G/H} be the set of cosets. Using the combinatorial model {\boldsymbol{\Omega}X\to\mathbf{P}X\to X} of the path fibration {{\Omega}Y\to{P}Y\to Y} and a canonical action {\mu\colon\boldsymbol{\Omega}X\times G/H\to G/H} , we construct a covering map {G/H\to Y_{H}\to Y} as the geometric realization of the associated short sequence {G/H\to\mathbf{P}X\times_{\mu}G/H\to X} . This construction, in particular, does not use the existence of a maximal tree in X. For a 2-dimensional X and {H=\{1\}} , it can also be viewed as a simplicial approximation of a Cayley 2-complex of G.