proposition: Lemma isKernel_opp {x y z : C^op} {f : C⟦x, y⟧} {g : C⟦y, z⟧} {Z : Zero C} {H : f · g = ZeroArrow Z _ _} (CK' : isCokernel Z f g H) {Z' : Zero C^op} : isKernel Z' (g : C^op⟦z, y⟧) (f : C^op⟦y, x⟧) (isCokernel_opp_eq f g Z H Z').
proof: Proof. set (CK := make_Cokernel _ _ _ _ CK'). use make_isKernel. - intros w h H'. rewrite <- (ZerosArrowEq C^op (Zero_opp Z) Z' w x) in H'. rewrite <- ZeroArrow_opp in H'. use unique_exists. + exact (CokernelOut Z CK w h H'). + use (CokernelCommutes Z CK). + intros y0. apply hs. + cbn. intros y0 X. use (CokernelOutsEq _ CK). rewrite (CokernelCommutes Z CK). cbn. rewrite X. apply idpath. 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