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Proper Functors and their Rational Fixed Point Milius, Stefan proper functor proper semiring coalgebra rational fixed point The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the final coalgebra. Inspired by Ésik and Maletti's notion of proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Stefan Milius 2017 Is Part Of LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CALCO.2017.18 urn:nbn:de:0030-drops-80314 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.18 eng https://creativecommons.org/licenses/by/3.0/legalcode