Bimorphisms of a pro*-category

Every morphism of an abstract coarse shape category Sh(C, D)* can be viewed as a morphism of the category pro*-D (defined on the class of inverse systems in D), where D is dense in C. Thus, the study of coarse shape isomorphisms reduces to the study of isomorphisms in the appropriate category pro*-D. In this paper bimorphisms in a category pro*-D are considered, for various categories D. We discuss in which cases pro*-D is a balanced category (category in which every bimorphism is an isomorphism). We are interested in the question whether the fact that one of the categories: D, pro-D and pro*-D is balanced implies that the other two categories are balanced. It is proved that if pro*-D is balanced then D is balanced. Further, if D admits sums and products and pro*-D is balanced then pro-D is balanced. In particular, pro*-C is balanced for C = Set (the category of sets and functions) and C = Grp (the category of groups and homomorphisms)