Lemma opp_isCokernel {x y z : C^op} {f : (C^op)⟦x, y⟧} {g : C^op⟦y, z⟧} {Z : Zero (C^op)} {H : f · g = ZeroArrow Z _ _} (K' : isKernel (C:=op_category C) Z f g H) {Z' : Zero C} : isCokernel Z' (g : C⟦z, y⟧) (f : C⟦y, x⟧) (opp_isCokernel_eq f g Z H Z'). Proof. set (K := make_Kernel _ _ _ _ K'). use make_isCokernel. - intros w h H'. rewrite <- (ZerosArrowEq C (opp_Zero Z) Z' z w) in H'. rewrite <- opp_ZeroArrow in H'. use unique_exists. + exact (KernelIn (C:=op_category _ )Z K w h H'). + use (KernelCommutes (C:=op_category _) Z K). + intros y0. apply hs. + cbn. intros y0 X. use (KernelInsEq (C:=op_category _) Z K). rewrite (KernelCommutes (C:=op_category _) Z K). cbn. rewrite X. apply idpath. 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.