Ultrametrization of pro*-morphism sets

For every pair of inverse systems $boldsymbol{X}$, $boldsymbol{Y}$ in a category $mathcal{A}$, where $boldsymbol{Y}$ is cofinite, there exists a complete ultrametric structure on the set $pro^{ast }mbox{-}mathcal{A}(boldsymbol{X},boldsymbol{Y})$. The corresponding hom-bifunctor is the internal and invariant $Hom$ of a subcategory, containing $tow^{ast }mbox{-}mathcal{A}$, in the category of complete metric spaces. Several applications to the shapes (ordinary, coarse and weak) are considered