Coinductive predicates express persisting ‘safety’ specifications of transition systems. Previous observations by Hermida and Jacobs identify coinductive predicates as suitable final coalgebras in a fibration – a categorical abstraction of predicate logic. In this paper, we follow the spirit of a seminal work by Worrell and study final sequences in a fibration. Our main contribution is to identify some categorical ‘size restriction’ axioms that guarantee stabilization of final sequences after ω steps. In its course, we develop a relevant categorical infrastructure that relates fibrations and locally presentable categories, a combination that does not seem to be studied a lot. The genericity of our fibrational framework can be exploited for binary relations (i.e. the logic of ‘binary predicates’) for which a coinductive predicate is bisimilarity, constructive logics (where interests are growing in coinductive predicates) and logics for name-passing processes.