proposition: Lemma isCokernel_opp {x y z : C} {f : C⟦x, y⟧} {g : C⟦y, z⟧} {Z : Zero C} {H : f · g = ZeroArrow Z _ _} (K' : isKernel Z f g H) {Z' : Zero C^op} : isCokernel Z' (g : C^op⟦z, y⟧) (f : C^op⟦y, x⟧) (isCokernel_opp_eq f g Z H Z').
proof: Proof. set (K := make_Kernel _ _ _ _ K'). use make_isCokernel. - intros w h H'. cbn in H'. set (XXX := (ZerosArrowEq C^op (Zero_opp Z) Z' z w)). use unique_exists. + use (KernelIn Z K w h _). rewrite ZeroArrow_opp. rewrite XXX. apply H'. + cbn. use (KernelCommutes Z K). + intros y0. apply (has_homsets_opp hs). + cbn. intros y0 X. use (KernelInsEq Z K). rewrite KernelCommutes. exact X. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts