proposition: Lemma ExactPushout' {M : ExactCategory} {A B C A':M} (i : A --> B) (p : B --> C) (pr : isExact2 i p) (r : A --> A') : ∃ (B':M) (i':A'-->B') (s:B-->B') (p':B'-->C), s·p' = p ∧ isPushout' (M:=M) i r s i' ∧ isExact2 i' p'.
proof: Proof. assert (I := ExactToAdmMono pr). assert (R := EC_PushoutMono i r I). use (hinhfun _ R); clear R; intros [po J]. set (pomap := PairPushoutMap (EC_ExactToKernelCokernel pr) r po). set (p' := pr1 pomap). exists po. exists (PushoutIn2 po). exists (PushoutIn1 po). exists p'. split. { exact (pr12 pomap). } split. { apply Pushout_to_isPushout'. } { use ExactSequenceFromMono. { exact (PairPushoutCokernel i p (EC_ExactToKernelCokernel pr) r po). } exact J. } 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