Axiomatization of the coherence property for categories of symmetries

Given an equational theory (Sigma, E), a relaxed (Sigma, E)-system is a category S enriched with a Sigma -algebra structure on both objects and arrows such that a natural isomorphism alpha (S) :t(S) double right arrow t'(S), called natural symmetry, exists for each t =(E) t'. A symmetry is an instance of a natural symmetry. A category of symmetries, which includes only symmetries, is a free object in the category of relaxed (Sigma, E)-systems. The coherence property states that the diagrams in a category of symmetries are commutative. In this paper we present a method for expressing the coherence property in an axiomatic way.