proposition: Lemma InvTranslationFunctorH_comm : functor_composite (ComplexHomotFunctor A) InvTranslationFunctorH = functor_composite InvTranslationFunctor (ComplexHomotFunctor A).
proof: Proof. use functor_eq. - apply to_has_homsets. - use functor_data_eq. + intros C. cbn. apply idpath. + intros C1 C2 f. unfold double_transport. rewrite idpath_transportf. rewrite idpath_transportf. assert (e1 : pr2 (pr1 (functor_composite (ComplexHomotFunctor A) InvTranslationFunctorH)) C1 C2 f = # InvTranslationFunctorH (# (ComplexHomotFunctor A) f)) by apply idpath. rewrite e1. clear e1. assert (e2 : pr2 (pr1 (functor_composite InvTranslationFunctor (ComplexHomotFunctor A))) C1 C2 f = # (ComplexHomotFunctor A) (# InvTranslationFunctor f)) by apply idpath. rewrite e2. clear e2. use (squash_to_prop (ComplexHomotFunctor_issurj A (# (ComplexHomotFunctor A) f))). { apply to_has_homsets. } intros f'. set (im := InvTranslationFunctorH_Mor_data (# (ComplexHomotFunctor A) f) f'). rewrite <- (@InvTranslationFunctorHImEq C1 C2 _ im f (idpath _)). use InvTranslationFunctorHImEq. exact (hfiberpr2 _ _ f'). 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.Core.Univalence 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