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

2362-cells. This means that a comonad (X, C) in C is a 1-cell C : X → X togetherwith 2-cells ∆ C : C → C · C and ε C : C → 1 X called comultiplication and counitsatisfying the reversed diagrams, i.e.(1C · ∆ C) ∆ C = ( )∆ C · 1 C ∆C(249)(1C · ε C) ∆ C = 1 C = ( )(250)ε C · 1 C ∆ C .A comonad functor is a pair (P, ψ) : (X, C) → (Y, D) where P : X → Y is a 1-cellin C and ψ : P · C → D · P is a 2-cell in C satisfying(εD · 1 P)ψ = 1P · ε C and (1 D · ψ) (ψ · 1 C ) ( 1 P · ∆ C) = ( ∆ D · 1 P)ψ.Finally, a comonad functorial morphism ω : (P ′ , ψ ′ ) → (P, ψ) is ω : P ′ → P is a2-cell in C satisfyingψ (ω · 1 C ) = (1 D · ω) ψ ′ .Now, we consider the category Cmd (C) ((X, 1 X ) , (Y, C)) where (X, 1 X ) and (Y, C)are 0-cells in Cmd (C) respectively with trivial comultiplication and counit theformer and ∆ C , ε C the latter. Note that C (X, C) : C (X, Y ) → C (X, Y ) is acomonad over the category C (X, Y ) with comultiplication and counit given byC ( X, ∆ C) = ∆ C () : C (X, C) → C (X, C · C) and C ( X, ε C) = ε C () : C (X, C) →C (X, 1 Y ). The objects of such category are the comonad functors (Q, ψ) : (X, 1 X ) →(Y, C) where Q : X → Y is a 1-cell and ψ : Q · 1 X → C · Q = C (X, C)Q is a 2-cellsatisfying ( ε C · 1 Q)ψ = 1Q and ( ∆ C · 1 Q)ψ = (1C · ψ) ψ so thatCmd (C) ((X, 1 X ) , (Y, C)) = C(X,C) C (X, Y ).Following the definition of the 2-category Mnd (C) for any bicategory C, we canconsider the 2-categories Mnd (Mnd (C)) and Mnd (BIM (C)) and the functorbetween themMnd (F (C)) : Mnd (Mnd (C)) → Mnd (BIM (C)) .Let us first consider Mnd (Mnd (C)):• 0-cells: pairs ((X, A) , (Q, φ)) where (X, A) is an object in Mnd (C) and(Q, φ) is a 1-cell in Mnd (C) together with a pair of 2-cells in Mnd (C) m (Q,φ)and u (Q,φ) satisfying associativity and unitality conditions. Therefore we havethat A : X → X is a 1-cell in C together with 2-cells m A = m (X,A) : A·A → Aand u A = u (X,A) : 1 X → A satisfying associativity and unitality conditionsand we have that Q : X → X is a 1-cell in C together with the 2-cell of Cφ : A · Q → Q · A satisfying the following conditions(251)(252)(253)(254)φ (m A · 1 Q ) = (1 Q · m A ) (φ · 1 A ) (1 A · φ)φ (u A · 1 Q ) = 1 Q · u AFinally, the 2-cells of Mnd (C)m (Q,φ) : (Q, φ) · (Q, φ) = (Q · Q, (1 Q · φ) (φ · 1 Q )) → (Q, φ) and u (Q,φ) :(1 X , 1 1X ) → (Q, φ) satisfying the associativity and unitality conditions, needsto satisfy also the followingφ ( 1 A · m (Q,φ))=(m(Q,φ) · 1 A)(1Q · φ) (φ · 1 Q )φ ( 1 A · u (Q,φ))= u(Q,φ) · 1 A .