An existence theorem concerning strong shape of Cartesian products

The paper is devoted to the question is the Cartesian product $Xtimes P$ of a compact Hausdorff space $X$ and a polyhedron $P$ a product in the strong shape category SSh of topological spaces. The question consists of two parts. The existence part, which asks whether, for a topological space $Z$, for a strong shape morphism $Fcolon Zto X$ and a homotopy class of mappings $[g]colon Zto P$, there exists a strong shape morphism $Hcolon Zto Xtimes P$, whose compositions with the canonical projections of $Xtimes P$ equal $F$ and $[g]$, respectively. The uniqueness part asks if $H$ is unique. The main result of the paper asserts that $H$ exists, whenever $Z$ is either metrizable or has the homotopy type of a polyhedron. If $X$ is a metric compactum, $H$ exists for all topological spaces $Z$. The proofs use resolutions of spaces and coherent homotopies of inverse systems. It is known that, in the ordinary shape category Sh, $H$ need not be unique, even in the case when $Z$ is a metrizable space or a polyhedron