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

194Following Theorem 6.29, we now calculate the monad(E, y) = Coequ Fun(z l , z r)where z l = (P χ) ◦ (δ D P Q) and z r = ε D P Q : DP Q → P Q. In our case, let usconsider∑ii∑ẑ l : (H ⊗ H) □ C H −→ H ⊗ H∑j ki,j ⊗ h i,j ⊗ g i ↦→ ∑ ij ki,j S ( h i,j) g i (1) ⊗ g i (2)and let us prove that ẑl it is left C-colinear. By (215) we have that ∑ iS (h i,j ) g i ∈ H ⊗ L so that, in view of (212), we have∑ ( ∑j π [ (k ) ) [ ( i,j(1) S hi,jg(1)]i ⊗ k ) ]i,j(2) S hi,jg(1)iHenceand= ∑ i∑j π (k i,j(1))(1)⊗ k i,j(2) S ( h i,j) g i (1) ⊗ g i (2).z l : −□ C H ⊗ H□ C H −→ −□ C H ⊗ H∑∑ ∑−□ C∑j ki,j ⊗ h i,j ⊗ g i ↦→ −□ Cii(2) ⊗ gi (3)j ki,j S ( h i,j) g i (1) ⊗ g i (2)z r : −□ C H ⊗ H□ C H −→ −□ C H ⊗ H∑∑−□ C∑j ki,j ⊗ h i,j ⊗ g i ↦→ −□ C∑j ki,j ⊗ h i,j ε ( H g i)iare well-defined. For every ( X, ρ C X)∈ Comod-C we haveso thatwhereI X□C z =E ( X, ρ C X)=X□ C H ⊗ HIm (X□ C z l − X□ C z r ) =E ( X, ρ C X)=X□ C H ⊗ HI X□C ziX□ C H ⊗ HIm (X□ C (z l − z r ))〈 ∑i,j xj ⊗ k j S (h i ) g i (1) ⊗ gi (2) − ∑ i,j xj ⊗ k j ⊗ h i ε H (g i )| ∑ j xj ⊗ k j , ∑ i hi ⊗ g i ∈ H□ C H∑j ki,j ⊗Recall (see [BrHaj, Theorem 3.5]) that, associated to the cocanonical map, we havea unique canonical entwining structure given byψ = (̂τ ⊗ H) ◦ ( H ⊗ ∆ H) ◦ cocan : H ⊗ L −→ L ⊗ Hh ⊗ y ↦→ y (1) ⊗ hy (2)where ̂τ = ( ε H ⊗ L ) ◦cocan −1 : H□ C H −→ L is the cotranslation map. Since cocanis an isomorphism, in order to understand better the monad E we first composewith the isomorphism H ⊗ cocan and we compute for every h, g ∈ H, y ∈ L,(z l ◦ (H ⊗ cocan) ) (h ⊗ g ⊗ y) = hy (1) ⊗ gy (2) and (z r ◦ (H ⊗ cocan)) (h ⊗ g ⊗ y) =〉.h ⊗ gε H (y). Leti : (H ⊗ H) L + → (H ⊗ H)