Axiomatic criteria are given for the existence of higher-order maps over subobjects and quotients. These criteria are applied in showing the soundness of a method for proving specification refinement up to observational equivalence. This generalises the method to handle data types with higher-order operations. We also give a direct setoid-based model satisfying the criteria. The setting is the second-order polymorphic lambda calculus and the assumption of relational parametricity.