A 'philosophical' interpretation of oriented graphs might be out of the scope of the categories mailing list, it seems elucidating to me though.(It could have been for Heraclit and Parmenides.) I suggest to consider vertices as states and edges as transitions. A reflexive edge corresponds to a transition from a state to itself. Two notions of morphism of oriented graphs come seem natural: - A rigid embedding which preserves difference of states - A notion of morphism which may equalize different states. The meaning of equality of states needs precision then. Coherence semantics deals with a different notion of graph. This is clear to me and can also be deduced from a discussion between Thomas Ehrhard and his student Pierre to which I assisted. This bears an answer to the question of reflexivity I think. Sincerely, Tom