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

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

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64and[(Ω ◦ Γ) (ϕ)] (Y ) = (LY, (ϕLY ) ◦ (LηY )) = Υ (ϕ) (Y ) and [(Ω ◦ Γ) (ϕ)] (f) = Lf.Remark 4.44. When C = LR = (LR, LηR, ɛ) and ϕ = Id LR the functor K =Υ (ϕ) : B → LR A such that LR U ◦ K = L is called the Eilenberg-Moore comparis<strong>on</strong>functor.Corollary 4.45. Let C = ( C, ∆ C , ε C) and D = ( D, ∆ D , ε D) be com<strong>on</strong>ads <strong>on</strong> acategory A. There exists a bijective corresp<strong>on</strong>dence between the following collecti<strong>on</strong>sof data:K Functors K : C A → D A such that D U ◦ K = C U,M com<strong>on</strong>ad morphisms ϕ : C → Dgiven byΨ : K → M where Ψ (K) = (Cɛ) ◦ ([D U ( γ D K )] C F )Υ : M → K where Υ (ϕ) (Y ) = (C UY, ( ϕ C UY ) ◦ (C Uγ C Y )) and Υ (ϕ) (f) = C U (f) .Proof. Apply Theorem 4.43 to the case L = C U : C A → A and R = C F : A → C Aand note that (LR, LηR, ɛ) = (C U C F , C Uγ C C F , ε C) = ( C, ∆ C , ε C) .□Propositi<strong>on</strong> 4.46. Let (L, R) be an adjuncti<strong>on</strong> where L : B → A and R : A →B , let C = ( C, ∆ C , ε C) be a com<strong>on</strong>ad <strong>on</strong> the category A and let ϕ : LR =(LR, LηR, ɛ) → C = ( C, ∆ C , ε C) be a com<strong>on</strong>ad morphism. Let α = Θ (ϕ) =(Rϕ) ◦ (ηR). Then the isomorphism a X,Y : Hom A (LY, X) → Hom B (Y, RX) of theadjuncti<strong>on</strong> (L, R) induces an isomorphismâ X,Y: HomC A(Kϕ Y, ( X, C ρ X))→ EquSets(HomB (Y, αX) , Hom B(Y, R C ρ X)).Proof. Leta X,Y : Hom A (LY, X) → Hom B (Y, RX)be the isomorphism of the adjuncti<strong>on</strong> (L, R) for every Y ∈ B and for every X ∈ A.Recall that a X,Y (ξ) = (Rξ) ◦ (ηY ) and a −1X,Y(ζ) = (ɛX) ◦ (Lζ) . Let us check thatwe can apply Lemma <str<strong>on</strong>g>2.</str<strong>on</strong>g>15 to the case Z = Hom A (L−, X) , Z ′ = Hom B (−, RX) ,W = Hom A (L−, CX) , W ′ = Hom( B (−, RCX), ) a = C ρ X ◦ −, b = C − ◦Γ (ϕ) Y,a ′ = Hom( B (−, αX), b ′ = Hom B −, R C ρ X and ϕ = aX,− , ψ = a CX,− , E =Equ CFun ρ X ◦ −, C (−) ◦ Γ (ϕ) Y ) (and( ))E ′ = Equ Fun HomB (−, αX) , Hom B −, R C ρ X(Equ CFun ρ X ◦ −, C (−) ◦ Γ (ϕ) − ) â ( ( ))X,− Equ Fun HomB (−, αX) , Hom B −, R C ρ X□iZ = Hom A (L−, X)a= C ρ X ◦−b=C−◦Γ(ϕ)−W = Hom A (L−, CX)a X,−a CX,−Z ′ = Hom B (−, RX)a ′ =Hom B (−,αX)i ′b ′ =Hom B(−,R C ρ X) W ′ = Hom B (−, RCX)

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