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

13such thatd 0 ◦ v = Id Yd 1 ◦ v = u ◦ cc ◦ u = Id Cc ◦ d 0 = c ◦ d 1 .Proposition <strong>2.</strong>24 ([BW, Proposition 2 (a)]). Let C be a category and let(C, X, Y, c, d 0 , d 1 , u, v) be a contractible coequalizer. Then (C, c) = Coequ C (d 0 , d 1 ) .Proof. Dual to Proposition <strong>2.</strong>24.Definition <strong>2.</strong>25. Let F : C → D be a functor. An F -contractible coequalizer pairis a parallel pairin C such that there exists a contractible coequalizerin D.<strong>2.</strong>3. Adjunction.F XXF d 0vF d 1d 0d 1<strong>2.</strong>26. Let L : B → A and R : A → B be functors. Recall that L is called a leftadjoint of R, or R is called a right adjoint of L if there exists functorial morphisms F Y Yη : Id B → RL and ɛ : LR → Id Asuch that(ɛL) ◦ (Lη) = L and (Rɛ) ◦ (ηR) = R.In this case we also say that (L, R) is an adjunction and η is called the unit of theadjunction while ɛ is called the counit of the adjunction. Leta X,Y : Hom A (LY, X) → Hom B (Y, RX)be the isomorphism of the adjunction (L, R). Then, for every ξ ∈ Hom A (LY, X)and for every ζ ∈ Hom B (Y, RX) we also havea X,Y (ξ) = (Rξ) ◦ (ηY ) and a −1X,Y(ζ) = (ɛX) ◦ (Lζ) .Moreover, for every X ∈ A, Y ∈ B, unit and counit of the adjunction are given byηY = a LY,Y (Id LY ) and ɛX = a −1X,RX (Id RX) .<strong>2.</strong>27. Let (L, R) be an adjunction. Then L preserves colimits and thus coequalizersand R preserves limits and thus equalizers. We also say that L is right exact andthat R is left exact.Lemma <strong>2.</strong>28. Let (L, R) be an adjunction with unit η and counit ɛ, where L : B → Aand R : A → B. For every Y ′ ∈ B the following conditions are equivalent:(1) L −,Y ′ = a −1LY ′ ,− ◦ Hom B (−, ηY ′ ) is a functorial isomorphismcuC□