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

In 2003 the author has associated with every cofinite 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 ( X , K ) of X × P , which consists of paracompact spaces. If X consists of compact polyhedra, then R ( X , K ) consists of spaces having the homotopy type of polyhedra. In a subsequent paper, published in 2007, the author proved that R ( X , K ) is a covariant functor in the first variable. In the present paper it is proved that R ( X , K ) is a covariant functor also in the second variable.