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

198Letandα l = (ϕ ⊗ H) ( (x ⊗ H) ◦ ( H□ C z l) ◦ (H□ C H ⊗ cocan) )α r = (ϕ ⊗ H) ((x ⊗ H) ◦ (H□ C z r ) ◦ (H□ C H ⊗ cocan)) .Then, for every ∑ h i ⊗ g i ∈ H□ C H, k ∈ H and t ∈ L,(∑ )(α l − α r ) h i ⊗ g i ⊗ k ⊗ t = ∑ S ( h i) g i t (1) ⊗ kt (2) − ∑ S ( h i) g i ⊗ kε H (t)[∑ (=) [∑ (S hig i ⊗ k]· t − ) ]S hig i ⊗ k · ε H (t) 1 L[∑ (=) S hig i ⊗ k]· (t )− ε H (t) 1 Lso that we getIm (α l − α r ) = (L ⊗ H) L +and hence the isomorphism ϕ : A = H□ CHI w−→ L induces an isomorphism( )( ( ̂Q, l = Coequ Fun (P x) ◦ z l P ) , (P x) ◦ (z r P ) )∼ = − ⊗ Coequ (αl , α r ) = − ⊗ L ⊗ H(L ⊗ H) L + .In the sequel, given elements l i ∈ L and h i ∈ H we will use the notation[∑ ]l i ⊗ h i = ∑ l i ⊗ h i + (L ⊗ H) L + .bQFollowing Proposition 7.6, the functor ̂Q can be equipped with the structure of aB-A-bimodule functor, i.e. in our setting, with a structure of E-L-bimodule functor.In particular E µb Q= − ⊗ ʵbQ: E ̂Q → ̂Q and µ L b Q= − ⊗ ̂µ L b Q: ̂QA → ̂Q where∑andi∑jE L ⊗ Hµb Q:(L ⊗ H) L □ H ⊗ H+ C(H ⊗ H) L −→ L ⊗ H+ (L ⊗ H) L +∑ [l i,j ⊗ k i,j] [Q b □ C h i,s ⊗ t i,s] ↦→ ∑ ∑ ∑ [E l i,j S ( k i,j) h i,s ⊗ t i,s] bi Qŝµ L b Q: L ⊗ L ⊗ H(L ⊗ H) L + −→ L ⊗ H(L ⊗ H) L +y ⊗ [y ′ ⊗ h]b Q↦→ [yy ′ ⊗ h]b Q.Such a bimodule functor ̂Q is the one giving rise, together with the functor Q, tothe equivalence of the categories of modules over the monads A ≃ L and B = Econstructed in the above Subsection 8.1 (in particular see Theorems 8.6 and 8.9).More explicitly,andLQ E = − ⊗ E H L : E B = E (Comod-C) → L A = L (Mod-k) = Mod-LE ̂Q L = − ⊗ LL ⊗ H(L ⊗ H) L + : LA = Mod-L → E B = E (Comod-C) .js