Terminal coalgebras

$$\nu F$$ of finitary endofunctors F on categories called strongly lfp are proved to carry a canonical ultrametric on their underlying sets. The subspace formed by the initial algebra $$\mu F$$ has the property that for every coalgebra A we obtain its unique homomorphism into $$\nu F$$ as a limit of a Cauchy sequence of morphisms into $$\mu F$$ called approximate homomorphisms. The concept of a strongly lfp category includes categories of sets, posets, vector spaces, boolean algebras, and many others.For the free completely iterative algebra $$\varPsi B$$ on a pointed object B we analogously present a canonical ultrametric on its underlying set. The subspace formed by the free algebra $$\varPhi B$$ on B has the property that for every recursive equation in $$\varPsi B$$ we obtain the unique solution as a limit of a Cauchy sequence of morphisms into $$\varPhi B$$ called approximate solutions. A completely analogous result holds for the free iterative algebra RB on B.