proposition: Lemma KAOctaComm4 {x y z : Complex A} (f1 : Morphism x y) (f2 : Morphism y z) : MorphismComp (KAOctaMor2 f1 f2) (MappingConePr1 A f2) = MorphismComp (MappingConePr1 A (MorphismComp f1 f2)) (# (TranslationFunctor A) f1).
proof: Proof. use MorphismEq. intros i. cbn. unfold TranslationMorphism_mor. set (DS1 := to_BinDirectSums A (x (i + 1)) (z i)). set (DS2 := to_BinDirectSums A (y (i + 1)) (z i)). rewrite to_postmor_linear'. rewrite <- assoc. rewrite <- assoc. rewrite (to_IdIn1 DS2). rewrite id_right. rewrite <- assoc. rewrite (to_Unel2' DS2). 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