proposition: Lemma TranslationHInvTranslationH_adjunction_eq1 (x : ComplexHomot_Additive A) : # TranslationFunctorH (# (ComplexHomotFunctor A) (TranslationTranslationInvNatTrans_Mor x)) · # (ComplexHomotFunctor A) (InvTranslationTranslationNatTrans_Mor (TranslationFunctorH x)) = identity (TranslationFunctorH x).
proof: Proof. use (pathscomp0 _ (functor_id TranslationFunctorH_AdditiveFunctor x)). set (tmp := functor_id (ComplexHomotFunctor A) x). apply (maponpaths (# TranslationFunctorH_AdditiveFunctor)) in tmp. use (pathscomp0 _ tmp). clear tmp. cbn. set (f' := @make_hfiber _ _ (# (ComplexHomotFunctor A)) (# (ComplexHomotFunctor A) (TranslationTranslationInvNatTrans_Mor x)) (TranslationTranslationInvNatTrans_Mor x) (idpath _)). set (tmp := TranslationFunctorH_Mor_Im _ f'). apply (maponpaths (postcompose (# (ComplexHomotFunctor A) (InvTranslationTranslationNatTrans_Mor (TranslationFunctorH x))))) in tmp. use (pathscomp0 tmp). clear tmp. unfold postcompose. set (id' := @make_hfiber _ _ (# (ComplexHomotFunctor A)) (# (ComplexHomotFunctor A) (identity _)) (identity (x : ob (ComplexPreCat_Additive A))) (idpath _)). set (tmp := TranslationFunctorH_Mor_Im _ id'). use (pathscomp0 _ (! tmp)). clear tmp. cbn. clear id'. set (tmp := functor_comp (ComplexHomotFunctor A) (# TranslationFunctor (TranslationTranslationInvNatTrans_Mor x)) (InvTranslationTranslationNatTrans_Mor (TranslationFunctorH x))). use (pathscomp0 (! tmp)). clear tmp. apply maponpaths. Local Transparent TranslationFunctorH. cbn. set (tmp := TranslationInvTranslation_adjunction_eq1 x). cbn in tmp. use (pathscomp0 tmp). clear tmp. apply pathsinv0. use (functor_id TranslationFunctor). 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