proposition: Lemma ShortShortExact_from_isCokernel_isKernel (SSED : ShortShortExactData A (to_Zero A)) (H : isCokernel (to_Zero A) (Mor1 SSED) (CokernelArrow (CoImage SSED)) (CoImage_Eq SSED)) : isKernel (to_Zero A) (KernelArrow (Image SSED)) (Mor2 SSED) (Image_Eq SSED).
proof: Proof. use make_isKernel. intros w h H'. use unique_exists. - apply (KernelIn (to_Zero A) (Image SSED) w h (ShortShortExact_from_isCokernel_eq1 SSED w h (ShortShortExact_from_isCokernel_eq2 SSED w h H' H) H)). - apply KernelCommutes. - intros y. apply hs. - intros y T. cbn in T. apply KernelInsEq. rewrite T. apply pathsinv0. apply KernelCommutes. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinDirectSums