Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

Since A U reflects and ( A Q BB P A , p A QBP A ) = Coequ Fun(µBQ P A , A Q B µ PA), there existsa unique functorial morphism AB σ A BA : A Q BB P A → Id A A such thatABσ A BA ◦ (p A QBP A ) = A σ A A.Using definition of σ A A , Bσ A B and Bσ A BA , naturality of λ B we compute(AU AB σ A BA)◦ (A Up A QBP A ) ◦ (Qp P ) = A U [ ABσ A BA ◦ (p A QBP A ) ] ◦ (Qp P )= ( AU A σ A A)◦ (QpP ) = ( A Uλ A ) ◦ ( σ A AU ) = ( A Uλ A ) ◦ ( Bσ A BAU ) ◦ (p QB P A U)(14)= ( A Uλ A ) ◦ ( Bσ A BAU ) ◦ (Q B λ BB P A U)= B σ A BA ◦ (Q B p B P ) ◦ (Q B λ BB P A U) = B σ A BA ◦ (Q B λ BB P A ) ◦ (Q BB F B Up B P )= B σ A BA ◦ (Q B λ BB P A ) ◦ (Qp P ) = ( AU AB σ A BA)◦ (A U A Q B λ BB P A ) ◦ (Qp P )and since Qp P is an epimorphism and A U reflects and by definition of AB σ A BA we getABσ A BA ◦ ( A Q B λ BB P A ) = AB σ A BA ◦ (p A QBP A ) = A σ A Aso that ɛ (A Q,P A ) = AB σ A BA ◦ ( AQ B λ BB P A ) = A σ A A .Lemma 6.23. Let M = (A, B, P, Q, σ A , σ B ) be a formal dual structure where theunderlying functors are A : A → A, B : B → B, P : A → B and Q : B → A.Assume that both categories A and B have coequalizers and the functors A, QBpreserve them. Assume that• C = ( C, ∆ C , ε C) is a comonad on the category A such that C preservescoequalizers• ˜C(= ˜C, ∆C e, ε e )Cis a lifting of the comonad C to the category A A( )• AQ, eC ρ A Q is a left ˜C-comodule functor• M is a tame Morita context.Then can 1 is an isomorphism if and only if A can A is an isomorphism if and onlyif A Q is a left ˜C-Galois functor.Proof. Assume that M is a tame Morita context. Then, by Corollary 6.22, ( A Q, P A )is an adjunction with counit ɛ := A σA A : AQP ( A → Id A ( A. Then, ) A Q is a left ˜C-Galois functor if and only if the morphism ˜CA σA)A eC◦ ρ A QP A = A can A is anisomorphism. By using Lemma 6.17 we deduce that can 1 is an isomorphism if andonly if A can A is an isomorphism if and only if A Q is a left ˜C-Galois functor. □The following Theorem is a formulation, in pure categorical terms, of [BV, Theorem<strong>2.</strong>18].Theorem 6.24. Let M = (A, B, P, Q, σ A , σ B ) be a regular tame Morita context.Assume that• both categories A and B have equalizers and coequalizers,• the functors A and B preserve equalizers,• the functors A, B, P, Q preserve coequalizers.Then the existence of the following structures are equivalent:(a) A herd τ : Q → QP Q for M101□