Metrization of pro-morphism sets

Every pair of inverse systems X, Y in a category A, where Y is cofinite, admits a complete (ultra)metric structure on the set pro-A(X,Y). The corresponding hom-bifunctor is not, generally, an internal Hom. However, there exists a subcategory of pro-A, containing tow-A, for which the hom-bifunctor is an invariant Hom into the category of complete metric spaces. Application to the sets tow-HcANR(X,Y) yields several new interesting results concerning Borsuk\u27s quasi-equivalence