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

Proposition 3.54. Let (L, R) be an adjunction where L : B → A and R : A → B,let A = (A, m A , u A ) be a monad on the category B and let ψ : A = (A, m A , u A ) →RL = (RL, RɛL, η) be a monad morphism. Let r = Θ (ψ) = (ɛL) ◦ (Lψ). Then thefunctor K ψ = Υ (ψ) : A → A B has a left adjoint D ψ : A B → A if and only if, forevery ( Y, A µ Y)∈ A B, there exists Coequ A(rY, L A µ Y). In this case, there exists afunctorial morphism d ψ : L A U → D ψ such thatand thus(D ψ , d ψ ) = Coequ Fun (r A U, L A Uλ A )[Dψ((Y, A µ Y)), dψ(Y, A µ Y)]= CoequA(rY, L A µ Y).Corollary 3.55. Let (L, R) be an adjunction where L : B → A and R : A →B. Let r = Θ (Id RL ) = ɛL. Then the functor K = Υ (Id RL ) : A → RL B has aleft adjoint D : RL B → A if and only, for every ( Y, RL µ Y)∈ RL B, there existsCoequ A(ɛLY, L RL µ Y). In this case, there exists a functorial morphism d : LRL U →D such that(D, d) = Coequ Fun (ɛL RL U, L RL Uλ RL )and thus[D((Y, RL µ Y)), d(Y, RL µ Y)]= CoequA(ɛLY, L RL µ Y).Remark 3.56. In the setting of Proposition 3.54, for every X ∈ A, we note thatthe counit of the adjunction (D ψ , K ψ ) is given by( )˜ɛX = ã −1X,K ψ X IdKψ X : Dψ K ψ (X) → X.We will consider the diagram( (( )) )(24) Hom A Dψ Y, A ã X,Y (( )µ Y , X Hom A B Y, A µ Y , Kψ X )41Hom A(d ψ((Y, A µ Y )),X)Hom A (LY, X)Hom A (rY,X)Hom A(L A µ Y ,X)Hom A (LAY, X)a X,Ya X,AYHom B (Y, RX)(Γ(ψ)X)◦(A−)Hom B( A µ Y ,RX) HomB (AY, RX)defining ã −,Y in the particular case of ( Y, A µ Y)= Kψ X. Note that, since K ψ X =(RX, (RɛX) ◦ (ψRX)) = (RX, lX) , we havei.e.(D ψ K ψ (X) , d ψ K ψ (X)) = (D ψ (RX, lX) , d ψ K ψ (X)) = Coequ B (rRX, LlX)= Coequ B ((ɛLRX) ◦ (LψRX) , (LRɛX) ◦ (LψRX))(25) (D ψ K ψ (X) , d ψ K ψ (X)) = Coequ B (rRX, LlX)where l = Γ (ψ) = (Rɛ) ◦ (ψR) . We compute(˜ɛX) ◦ (d ψ K ψ X) = Hom A (d ψ K ψ X, X) ((˜ɛX))( (= Hom A (d ψ K ψ X, X) ã −1X,K ψ X IdKψ X) )