Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P = | K | a resolution R K ( X ) of X × P , which consists of paracompact spaces. If X consists of compact polyhedra, then R K ( X ) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X × P . Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X ′ , such that X is strong shape dominated by X ′ and the Cartesian product X ′ × P is a direct product in Sh(Top), then also X × P is a direct product in the shape category Sh(Top).