proposition: Lemma make_ShortShortExact_isKernel_isKernel (SSED : ShortShortExactData A (to_Zero A)) (H : isKernel (to_Zero A) (Mor1 SSED) (Mor2 SSED) (ShortShortExactData_Eq (to_Zero A) SSED)) : isKernel (to_Zero A) (KernelArrow (Image SSED)) (Mor2 SSED) (Image_Eq SSED).
proof: Proof. set (K := make_Kernel _ _ _ _ H). set (e1 := factorization1 (Mor1 SSED)). cbn in e1. unfold Image. assert (e : is_z_isomorphism (CokernelArrow (Abelian.CoImage (Mor1 SSED)) · CoIm_to_Im (Mor1 SSED))). { use monic_epi_is_iso. - use isMonic_postcomp. + exact (Ob2 SSED). + exact (KernelArrow (Abelian.Image (Mor1 SSED))). + rewrite <- e1. use (KernelArrowisMonic (to_Zero A) K). - exact (factorization1_is_epi (Mor1 SSED)). } use Kernel_up_to_iso_isKernel. + exact K. + exact (z_iso_inv (make_z_iso _ _ e)). + apply (maponpaths (λ g : _, (inv_from_z_iso (make_z_iso _ _ e)) · g)) in e1. use (pathscomp0 _ (! e1)). clear e1. rewrite assoc. cbn. rewrite (is_inverse_in_precat2 e). rewrite id_left. apply idpath. 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