From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive orderunit spaces and reflexive basenorm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the norestriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach orderunit spaces or all Banach basenorm spaces, but not both at the same time.
