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

e a L-contractible equalizer pair in B. Then (53) has an equalizer d : X ′′ → X ′ inB andLX ′′ Ld LX ′ Ld 0 LXLd 1is an equalizer in A.Proof. Since L is a ϕ-comonadic functor we know that K ϕ = Υ (ϕ) : B → C A is anequivalence of categories. Then, instead of consideringin the category B, we can considerX ′ d 0 Xd 1K ϕ X ′ K ϕd 0 K ϕ XK ϕd 1in C A which is a C U-contractible equalizer pair. Let us denote by ( Z ′ , C ρ Z ′):= Kϕ X ′and ( )Z, C ρ Z := Kϕ X so that we can rewrite the C U-contractible equalizer pair asfollows(Z ′ , C K ϕd 0ρ Z ′) ( )Z, C ρ ZK ϕd 1We want to prove that this pair has an equalizer in C A. Since the pair (K ϕ d 0 , K ϕ d 1 )is a C U-contractible equalizer in C A, we have thatZ ′′ d Z ′sis a contractible equalizer and thus, by Proposition <strong>2.</strong>19, an equalizer in A. Let usconsider the following diagramC ρ Z ′′C C ρ Z ′′Ld 0tLd 1Z ′′ d Z ′C ρ Z ′CZ ′′ Cd CZ ′ ∆ C Z ′∆ C Z ′′C C ρ Z ′ ZLd 0Ld 1CLd 0CLd 1CCZ ′′ CCd CCZ ′ CCLd 0 CCLd 1 Z CZC C ρ ZC ρ Z CCZBy Proposition <strong>2.</strong>20, all the rows are contractible equalizers. Since Ld 0 = C UK ϕ d 0and Ld 1 = C UK ϕ d 1 where K ϕ d 0 and K ϕ d 1 are morphisms in C A, we have that theupper right square serially commutes. Moreover, since we also have that ∆ C is afunctorial morphism, the lower right square serially commutes. We also have thatC ρ Z ′◦d is a fork for (CLd 0 , CLd 1 ) and, since (CZ ′′ , CZ ′ , CZ, Cd, CLd 0 , CLd 1 , Cs, Ct)is a contractible equalizer, in particular (CZ ′′ , Cd) = Equ A (CLd 0 , CLd 1 ); by theuniversal property of the equalizer, there exists a unique morphism C ρ Z ′′ : Z ′′ →CZ ′′ such that(54)C ρ Z ′ ◦ d = (Cd) ◦ C ρ Z ′′.∆ C Z75