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On Terminal Coalgebras Derived from Initial Algebras Adámek, Jiří terminal coalgebras initial algebras algebraically complete category finitary functor fixed points of functors A number of important set functors have countable initial algebras, but terminal coalgebras are uncountable or even non-existent. We prove that the countable cardinality is an anomaly: every set functor with an initial algebra of a finite or uncountable regular cardinality has a terminal coalgebra of the same cardinality. We also present a number of categories that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Finally, for finitary set functors we prove that the initial algebra mu F and terminal coalgebra nu F carry a canonical ultrametric with the joint Cauchy completion. And the algebra structure of mu F determines, by extending its inverse continuously, the coalgebra structure of nu F. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jiří Adámek 2019 Is Part Of LIPIcs, Volume 139, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CALCO.2019.12 urn:nbn:de:0030-drops-114403 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2019.12 eng https://creativecommons.org/licenses/by/3.0/legalcode