The Cartesian product of a compactum and a space is a bifunctor in shape

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 two subsequent papers the author proved that R ( X , K ) is a covariant functor in each of its variables X and K. In the present paper it is proved that R ( X , K ) is a bifunctor. Using this result, it is proved that the Cartesian product X × Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh ( Cpt ) × Sh ( Top ) → Sh ( Top ) from the product category of the strong shape category of compact Hausdorff spaces SSh ( Cpt ) and the shape category Sh ( Top ) of topological spaces to the category Sh ( Top ) . This holds in spite of the fact that X × Z need not be a direct product in Sh ( Top ) .