A duality theory is a theory with 3 types E, E' and K (or just both types E, E' when taking for K the pair of booleans); and at least the following structures and axioms:
∀f∈F^{E},∀g, g_{'}∈E'^{F'},(∀x∈E, ∀y∈F',〈x,g(y)〉 = 〈f(x),y〉' = 〈x,g'(y)〉) ⇒ (∀y∈F', g(y) = g'(y)) ⇒ g=g'Based on this uniqueness, we shall say that g is the transpose of f, and written g = ^{t}f.
As (∏_{y}_{∈}_{F'} y) is an R-embedding from F to K^{F'}, we haveNow, let S ⊂ Pol R ⊂ Op_{K}. As seen above, in any K-duality system (E,E') we have E* ∈ Sub_{S}(K^{E}).
As E'⊂E*, the condition f∈Mor((E,E'),(F,F')), that is ∀y∈F', y०f ∈ E', is more restrictive, thus the inclusion.
∀f∈F^{E}, f ∈ Mor_{R}(E,F) ⇔ (∏_{y}_{∈}_{F'} y)०f ∈ Mor_{R}(E,K^{F'})
⇔ ∀y∈F', y०f ∈ Mor_{R}(E,K)=E*
⇔ f ∈ Mor((E,E*),(F,F'))