Lemma TranslationHInvTranslationH_adjunction_eq1 (x : ComplexHomot_Additive A) : # TranslationFunctorH (# (ComplexHomotFunctor A) (TranslationTranslationInvNatTrans_Mor x)) · # (ComplexHomotFunctor A) (InvTranslationTranslationNatTrans_Mor (TranslationFunctorH x)) = identity (TranslationFunctorH x). 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.
Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Abelian. Require Import UniMath.CategoryTheory.AbelianToAdditive. Require Import UniMath.CategoryTheory.AdditiveFunctors. Require Import UniMath.HomologicalAlgebra.Complexes. Require Import UniMath.HomologicalAlgebra.KA.