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In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X P is a product in Sh(Top), for every polyhedron P.