On some products in pro-categories and shape categories

This paper has two parts. In the first part we present a case in which we can calculate in a simple way the product of two inverse systems as object in a pro-category of a category with directed products, without counting systems at the same set of indices. In the second part, which is related to some papers of Keesling [4], Kodama [5], Mardešić [7], and Dydak-Mardešić [10], we give a sufficient condition by which a directed product of a category 𝒯 is transformed by a shape functor S : 𝒯 → Sh(𝒯, 𝒫) into a directed product. This theorem is applied to the toplogical shape category Sh(Top) where some examples are obtained.