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

34where A Q : B → A A is the functor defined in Lemma 3.29.Proof. Let ( Q : B → A, A µ Q)be a left A-module functor. Then, by Lemma 3.29,there exists a unique functor A Q : B → A A such thatAU ◦ A Q = Q and A Uλ AA Q = A µ Q .Note that, since AQ preserves coequalizers, by Lemma 3.19, Q = A U ◦ A Q preservescoequalizers. Then, by Lemma 3.20, also A Q preserves coequalizers. Conversely, ifH : B → A A is a functor preserving coequalizers, we get that A U ◦ H : B → A.Moreover, by Lemma 3.21, A U preserves coequalizers and thus also A U ◦H preservescoequalizers. Now, let us prove that A ν and A κ determine a bijective correspondencebetween A F and ( A A ← B). We compute( A κ ◦ A ν) (( Q, A µ Q))= A κ ( A Q) = ( A U A Q, A Uλ AA Q) = ( Q, A µ Q).On the other hand we have( A ν ◦ A κ) (H) = A ν (( A U ◦ H, A Uλ A H)) = A ( A U ◦ H) (17)= H.Theorem 3.39. Let A = (A, m A , u A ) be a monad on a category A with coequalizerssuch that A preserves coequalizers. Let B = (B, m B , u B ) be a monad on a categoryB with coequalizers such that B preserves coequalizers. Then there exists a bijectivecorrespondence between the following collections of data:AF B A-B-bimodule functors Q : B → A such that AQ and QB preserve coequalizers( A A ← B B) functors G : B B → A A preserving coequalizersgiven byAν B : AF B → ( A A ← B B) where A ν B((Q, A µ Q , µ B Q))= A Q BAκ B : ( A A ← B B) → A F B where A κ B (G) = ( A U ◦ G ◦ B F , A Uλ A G B F , A UGλ BB F ) .Proof. Let us consider an A-B-bimodule functor ( Q : B → A, A µ Q , µ Q) B such that AQand QB preserve coequalizers. In particular, ( Q, µ Q) B is a right B-module functor,so that we (( can apply )) the map ν B : F B → (A ← B B) of Theorem 3.37 and we get afunctor ν B Q, µBQ = QB : B B → A which preserves coequalizers. By Proposition3.30, ( )Q B , A µ QB is a left A-module functor so that we can also apply the mapAν : A F → ( A A ← B) of Theorem 3.38 where the category B is B B. The map A νis defined by A ν (( ))Q B , A µ QB = A (Q B ) = A Q B : B B → A A and A Q B preservescoequalizers. Conversely, let us consider a functor G : B B → A A which preservescoequalizers. By Theorem 3.38, we get a left A-module functor given byAκ (G) = ( A U ◦ G, A Uλ A G)where A U ◦ G : B B → A and A A UG preserves coequalizers. By Lemma 3.19, alsoAU ◦ G : B B → A preserves coequalizers. Thus, we can apply Theorem 3.37 and weget a right B-module functorκ B ( A UG) = ( A UG B F, A UGλ BB F )□