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

250be the codiagonal map of the ρ i . Then ρ M is uniquely defined byρ M ◦ ε i = ρ i .Then ( M, ρ M) ∈ (Mod-A) C . In fact, for every i ∈ I we have(ρ M ⊗ A C ) ◦ ρ M ◦ ε i = ( ρ M ⊗ A C ) ◦ ρ i = ( ρ M ⊗ A C ) ◦ (ε i ⊗ A C) ◦ ρ i= (ρ i ⊗ A C) ◦ ρ i = (ε i ⊗ A C) ◦ (ρ i ⊗ A C) ◦ ρ i = (ε i ⊗ A C) ◦ ( M i ⊗ A ∆ C) ◦ ρ iand= ( M ⊗ A ∆ C) ◦ (ε i ⊗ A C) ◦ ρ i = ( M ⊗ A ∆ C) ◦ ρ i = ( M ⊗ A ∆ C) ◦ ρ M ◦ ε ir M ◦ ( M ⊗ A ε C) ◦ ρ M ◦ ε i = r M ◦ ( M ⊗ A ε C) ◦ ρ i = r M ◦ ( M ⊗ A ε C) ◦ (ε i ⊗ A C) ◦ ρ i= r M ◦ (ε i ⊗ A A) ◦ ( M i ⊗ A ε C) ◦ ρ i = ε i ◦ r Mi ◦ ( M i ⊗ A ε C) ◦ ρ i = ε i .Let f : ( M, ρ M) → ( N, ρ N) be a morphism in (Mod-A) C so that f : M → N is amorphism in Mod-A and let us considerMf−→ Np−→ Coker (f) → 0the cokernel of f in Mod-A. Then we have the following diagram in Mod-AMfNpCoker (f)0We computeρ Mρ Nρ Coker(f)M ⊗ A C f⊗ AC N ⊗ A C p⊗ AC Coker (f) ⊗A C 0(p ⊗ A C) ◦ ρ N ◦ f = (p ⊗ A C) ◦ (f ⊗ A C) ◦ ρ M = (pf ⊗ A C) ◦ ρ M = 0by the universal property of the cokernel there exists a unique morphism ρ Coker(f) :Coker (f) → Coker (f) ⊗ A C such thatρ Coker(f) ◦ p = (p ⊗ A C) ◦ ρ N .Let us check that ( Coker (f) , ρ Coker(f)) ∈ (Mod-A) C . Let us compute(ρ Coker(f) ⊗ A C ) ◦ ρ Coker(f) ◦ p = ( ρ Coker(f) ⊗ A C ) ◦ (p ⊗ A C) ◦ ρ Nand= (p ⊗ A C ⊗ A C) ◦ ( ρ N ⊗ A C ) ◦ ρ N = (p ⊗ A C ⊗ A C) ◦ ( N ⊗ A ∆ C) ◦ ρ N= ( Coker (f) ⊗ A ∆ C) ◦ (p ⊗ A C) ◦ ρ N = ( Coker (f) ⊗ A ∆ C) ◦ ρ Coker(f) ◦ pr Coker(f) ◦ ( Coker (f) ⊗ A ε C) ◦ ρ Coker(f) ◦ p= r Coker(f) ◦ ( Coker (f) ⊗ A ε C) ◦ (p ⊗ A C) ◦ ρ N= r Coker(f) ◦ (p ⊗ A A) ◦ ( N ⊗ A ε C) ◦ ρ N = p ◦ r N ◦ ( N ⊗ A ε C) ◦ ρ N = pand since p is epi we conclude. Now, let ζ : ( N, ρ N) → ( Z, ρ Z) be a morphism in(Mod-A) C such that ζ◦f = 0. Then, there exists a unique morphism ζ ′ : Coker (f) →Z in Mod-A such that ζ ′ ◦p = ζ. We want to prove that ζ ′ ∈ (Mod-A) C . We computeρ Z ◦ ζ ′ ◦ p = ρ Z ◦ ζ = (ζ ⊗ A C) ◦ ρ N