Lemma Cokernel_opp_isKernel {y z : C} (f : C⟦y, z⟧) (Z : Zero C) (CK : @Cokernel C Z y z f) : isKernel (Zero_opp Z) (CokernelArrow CK) f (Cokernel_opp_eq f Z CK). Proof. use make_isKernel. - intros w h H'. cbn in H'. use unique_exists. + rewrite <- ZeroArrow_opp in H'. exact (CokernelOut Z CK w h H'). + cbn. use CokernelCommutes. + intros y0. apply (has_homsets_opp hs). + cbn. intros y0 X. use CokernelOutsEq. rewrite CokernelCommutes. exact X. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. @ZeroArrowTo (C^op) Z x = @ZeroArrowFrom C (opp_Zero Z) x. Lemma opp_ZeroArrowFrom {x : C} (Z : @Zero (C^op)) : @ZeroArrowFrom (C^op) Z x = @ZeroArrowTo C (opp_Zero Z) x. apply ArrowsFromZero. rewrite opp_ZeroArrowFrom. @ZeroArrowTo C Z x = @ZeroArrowFrom (C^op) (Zero_opp Z) x. Lemma ZeroArrowFrom_opp {x : C} (Z : @Zero C) : @ZeroArrowFrom C Z x = @ZeroArrowTo (C^op) (Zero_opp Z) x. apply ArrowsFromZero. rewrite ZeroArrowFrom_opp.