Categorical studies of recursive data structures and their associated reasoning principles have mostly focused on two extremes: initial algebras and induction, and final coalgebras and coinduction. In this paper we study their in-betweens. We formalize notions of alternating fixed points of functors using constructions that are similar to that of free monads. We find their use in categorical modeling of accepting run trees under the Büchi and parity acceptance condition. This modeling abstracts away from states of an automaton; it can thus be thought of as the “behaviors” of systems with the Büchi or parity conditions, in a way that follows the tradition of coalgebraic modeling of system behaviors.