Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we present an abstract categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs—but our focus is on the structural aspect of fibrational frameworks. Specifically we use morphisms of fibrations as well as canonical liftings of functors via Kan extensions. We apply this categorical framework by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.