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

and since A preserves equalizers, Ae is a monomorphism, so that we get(m A E) ◦ (Az ′ ) = z ′ ◦ A µ Z .Lemma 3.23. Let A = (A, m A , u A ) be a monad over a category A, let L, M : B → Abe functors and let µ : AL → L be an associative and unital functorial morphism,that is (L, µ) is a left A-module functor. Let h : L → M and let ϕ : AM → M befunctorial morphisms such that(5) h ◦ µ = ϕ ◦ (Ah) .If AAh and h are epimorphisms, then ϕ is associative and unital, that is (M, ϕ) isa left A-module functor.Proof. We calculateϕ ◦ (Aϕ) ◦ (AAh) (5)= ϕ ◦ (Ah) ◦ (Aµ) (5)= h ◦ µ ◦ (Aµ)µis ass.= h ◦ µ ◦ (m A L) (5)= ϕ ◦ (Ah) ◦ (m A L) m A= ϕ ◦ (m A M) ◦ (AAh) .Since AAh is an epimorphism, we deduce that ϕ is associative. Moreover we haveϕ ◦ (u A M) ◦ h u A= ϕ ◦ (Ah) ◦ (u A L) (5)= h ◦ µ ◦ (u A L) = h.Since h is an epimorphism, we get that ϕ is unital.3.<strong>1.</strong> Liftings of module functors.Proposition 3.24 ([Ap] and [J]). Let A = (A, m A , u A ) be a monad on a categoryA and let B = (B, m B , u B ) be a monad on a category B and let Q : A → B be afunctor. Then there is a bijection between the following collections of dataF functors ˜Q : A A → B B that are liftings of Q (i.e. B U ˜Q = Q A U)M functorial morphisms Φ : BQ → QA such thatΦ ◦ (m B Q) = (Qm A ) ◦ (ΦA) ◦ (BΦ) and Φ ◦ (u B Q) = Qu Agiven by( )a : F → M where a ˜Q =( )BUλ B ˜QA F ◦ ( B U B F Qu A )b : M → F where B Ub (Φ) = Q A U and B Uλ B b (Φ) = (Q A Uλ A ) ◦ Φ i.e.b : M → F where b (Φ) (( X, A µ X))=(QX,(Q A µ X)◦ (ΦX))and b (Φ) (f) = Q (f) .Proof. Let ˜Q : A A → B B be a lifting of the functor Q : A → B (i.e. B U ˜Q = Q A U).Define a functorial morphism φ : B F Q → ˜Q A F as the composite( )φ := λ B ˜QA F ◦ ( B F Qu A )where u A : A → A U A F = A is also the unit of the adjunction ( A F, A U) and λ B :BF B U → B B is the counit of the adjunction. Let now defineΦ def= B Uφ : B U B F Q = BQ → B U ˜Q A F = Q A U A F = QA.23□□