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

Theorem 3.37. Let B = (B, m B , u B ) be a monad on a category B with coequalizerssuch that B preserves coequalizers. Then there exists a bijective correspondencebetween the following collections of data:F B right B-module functors Q : B → A such that QB preserves coequalizers(A ← B B) functors G : B B → A preserving coequalizersgiven byν B : F B → (A ← B B) where ν B((Q, µBQ))= QBκ B : (A ← B B) → F B where κ B (G) = (G B F,Gλ BB F )where Q B is uniquely determined by (Q B , p Q ) = Coequ Fun(µBQ BU, Q B Uλ B).Proof. Let Q : B → A be a right B-module functor. Then we can consider Q B :BB → A defined by (6) as(Q B , p Q ) = Coequ Fun(µBQ BU, Q B Uλ B).Since by assumption QB preserves coequalizers, by Lemma 3.19 also Q preservescoequalizers. Moreover, since B preserves coequalizers, by Lemma 3.21 also thefunctor B U preserves coequalizers. Thus both QB B U and Q B U preserve coequalizers.By Corollary <strong>2.</strong>12 we get that also Q B : B B → A preserves coequalizers.Conversely, let us consider a functor G : B B → A that preserves coequalizers. ByProposition 3.35 we can consider the right B-module functor defined as followsQ = G ◦ B F and let µ B Q = Gλ BB F .Since B F is left adjoint to B U, in particular B F preserves coequalizers and sinceby assumption G preserves coequalizers, we get that also Q = G ◦ B F preservescoequalizers and so does QB.Now, we want to prove that ν B and κ B determine a bijective correspondence betweenF B and (A ← B B). Let us start with a right B-module functor ( Q : B → A, µ B Q).Then we haveMoreover we have(κ B ◦ ν B ) (( Q, µ Q)) B = κB (Q B ) = (Q BB F,Q B λ BB F )= ( )Q BB F,µ B (15)Q B B F = ( Q,µ Q) B .(ν B ◦ κ B ) (G) = ν B ((G B F,Gλ BB F )) = (G B F ) B(16)= G.Theorem 3.38. Let A = (A, m A , u A ) be a monad on a category A with coequalizerssuch that A preserves coequalizers. Then there exists a bijective correspondencebetween the following collections of data:AF left A-module functors Q : B → A such that AQ preserves coequalizers( A A ← B) functors H : B → A A preserving coequalizersgiven byAν : AF → ( A A ← B) where A ν (( Q, A µ Q))= A QAκ : ( A A ← B) → A F where A κ (H) = ( A U ◦ H, A Uλ A H)33□