proposition: Lemma KAOctaComm3 {x y z : Complex A} (f1 : Morphism x y) (f2 : Morphism y z) : MorphismComp (KAOctaMor3 f1 f2) (MappingConePr1 A (KAOctaMor1 f1 f2)) = MorphismComp (MappingConePr1 A f2) (# (TranslationFunctor A) (MappingConeIn2 A f1)).
proof: Proof. use MorphismEq. intros i. cbn. set (DS1 := to_BinDirectSums A (y (i + 1)) (z i)). set (DS2 := to_BinDirectSums A (x (i + 1 + 1)) (y (i + 1))). set (DS3 := to_BinDirectSums A (x (i + 1)) (z i)). set (DS4 := to_BinDirectSums A DS2 DS3). rewrite to_postmor_linear'. rewrite <- (assoc _ _ (to_Pr1 DS4)). rewrite <- (assoc _ _ (to_Pr1 DS4)). rewrite (to_IdIn1 DS4). rewrite id_right. rewrite (to_Unel2' DS4). rewrite ZeroArrow_comp_right. rewrite to_runax''. apply idpath. Qed.
imports: UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.NumberSystems.Integers UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts 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.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.AdditiveFunctors UniMath.HomologicalAlgebra.Complexes UniMath.HomologicalAlgebra.KA UniMath.HomologicalAlgebra.TranslationFunctors