Comparing monomorphisms and epimorphisms in pro and -categories

Given a category pair ( C , D ) , where D is dense in C , the abstract coarse shape category Sh ( C , D ) * was recently founded. It is realized via the category pro * - D defined on the class of all inverse systems in D . In this paper monomorphisms and epimorphisms in the category pro * - C are considered, for various categories C . The characterizations of epimorphisms (monomorphisms) in the category pro * - C are given, provided C admits products (sums). Since, one may consider the category pro - C as a subcategory of pro * - C , we discuss in which cases an epimorphism (monomorphism) in pro - C is an epimorphism (monomorphism) in pro * - C as well. We answered this question affirmatively for a category C admitting products (sums). It is shown by examples that the answer is generally negative, i.e. there exists a certain category C and an epimorphism (monomorphism) in pro - C which is not an epimorphism (monomorphism) in pro * - C .