Reflexive graphs and parametric polymorphism

The pioneering work on relational parametricity for the second order lambda calculus was done by John Reynolds under the assumption of the existence of set-based models, and subsequently reformulated by him, in conjunction with his student Ma, using the technology of PL-categories. The aim of this paper is to use the different technology of internal category theory to re-examine Ma and Reynolds' definitions. Apart from clarifying some of their constructions, this view enables us to prove that if we start with a non-parametric model which is left exact and which satisfies a completeness condition corresponding to Ma and Reynolds `suitability for polymorphism', then we can recover a parametric model with the same category of closed types. This implies, for example, that any suitably complete model (such as the PER model) has a parametric counterpart.