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.
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