proposition: Lemma AdditiveCoequalizerToCokernel_isCokernel {x y : ob A} (f g : x --> y) (CE : Coequalizer f g) : isCokernel (to_Zero A) (to_binop x y f (to_inv g)) (CoequalizerArrow CE) (AdditiveCoequalizerToCokernel_eq1 f g CE).
proof: Proof. use make_isCokernel. - intros w h H'. use unique_exists. + use CoequalizerOut. * exact h. * use (to_rcan A (to_inv g · h)). rewrite <- to_postmor_linear'. rewrite <- to_postmor_linear'. rewrite (@to_rinvax' A (to_Zero A)). rewrite ZeroArrow_comp_left. exact H'. + cbn. use CoequalizerCommutes. + intros y0. apply to_has_homsets. + intros y0 X. cbn in X. use (CoequalizerArrowisEpi CE). rewrite CoequalizerCommutes. exact X. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.MoreFoundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.Monoids UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinDirectSums