In the context of formal verification in general and model checking in particular, \emph{parity games} nowadays serve as a mighty vehicle: many problems are encoded as a parity game, which is then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of \emph{progress measure}, and formalize it in general, possibly infinitary, lattice-theoretic terms. We then apply it to a general model-checking framework where systems are categorically presented as coalgebras, whose theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems.