proposition: Lemma IsomMono1 {M : ExactCategory} {A B B':M} (f : A --> B) (f' : A --> B') : IsoArrowFrom f f' -> isAdmissibleMonomorphism f -> isAdmissibleMonomorphism f'.
proof: Proof. intros [i I] E. apply (squash_to_hProp E); clear E; intros [C [p E]]. apply hinhpr. exists C. exists (z_iso_inv i · p). use (EC_IsomorphicToExact _ E). exists (identity_z_iso A). exists i. exists (identity_z_iso C). split; cbn. - split. + exact (id_left _ @ ! I). + exact (I @ ! id_left _). - split. + refine (assoc _ _ _ @ _ @ id_left _ @ ! id_right _). apply (maponpaths (λ k, k · p)). apply z_iso_inv_after_z_iso. + apply pathsinv0. refine (assoc _ _ _ @ _ @ id_left _ @ ! id_right _). apply (maponpaths (λ k, k · p)). apply z_iso_inv_after_z_iso. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.Notations UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Groups UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Subcategory.Full UniMath.MoreFoundations.Propositions