  Despite its success in producing numerous general results on state-based
dynamics, the theory of coalgebra has struggled to accommodate the Buechi
acceptance condition---a basic notion in the theory of automata for infinite
words or trees. In this paper we present a clean answer to the question that
builds on the "maximality" characterization of infinite traces (by Jacobs and
Cirstea): the accepted language of a Buechi automaton is characterized by two
commuting diagrams, one for a least homomorphism and the other for a greatest,
much like in a system of (least and greatest) fixed-point equations. This
characterization works uniformly for the nondeterministic branching and the
probabilistic one; and for words and trees alike. We present our results in
terms of the parity acceptance condition that generalizes Buechi's.
