一句话：引入 proper functor，证明其理性不动点是终结余代数的子余代数，用于余代数正则表达式演算的可靠性/完备性。
本文研究代数范畴上集合函子的理性不动点，它通常刻画有限余代数的行为，但在代数范畴上可能不再是完全抽象的，即未必是终结余代数的子余代数。受 Ésik 与 Maletti 的 proper semiring 概念启发，作者引入 proper functor。主要结果是：对 proper 函子，理性不动点可表示为所有以自由有限生成代数为载体对象的余代数的余极限，并且是终结余代数的子余代数；同时证明该余极限是终结余代数的子余代数当且仅当函子是 proper 的。这些结果为余代数正则表达式演算（如加权自动机）的可靠性与完备性证明提供技术工具。
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 fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a 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.