Proposition duality_involution_inv_of_2cell_psfunctor_eq {x y : B} {f g : x --> y} (τ : g ==> f) : duality_involution_inv_of_2cell (## L τ) = τ. Proof. unfold duality_involution_inv_of_2cell. simpl. rewrite !vassocl. use vcomp_move_R_pM ; [ is_iso | ]. simpl. rewrite <- vcomp_runitor. rewrite !vassocr. apply maponpaths_2. rewrite !vassocl. etrans. { do 2 apply maponpaths. rewrite !vassocr. do 2 apply maponpaths_2. rewrite !rwhisker_vcomp. apply maponpaths. rewrite !vassocl. etrans. { apply maponpaths. exact (psnaturality_natural η _ _ _ _ τ). } rewrite !vassocr. rewrite vcomp_linv. apply id2_left. } etrans. { apply maponpaths. rewrite !vassocr. rewrite rwhisker_rwhisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. rewrite lassociator_rassociator. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite <- vcomp_whisker. rewrite !vassocl. rewrite lwhisker_vcomp. refine (_ @ id2_right _). apply maponpaths. rewrite <- lwhisker_id2. apply maponpaths. exact (vcomp_rinv (invertible_modcomponent_of (unit_unit_inv_of_duality HL) y)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.Properties.ContainsAdjEquiv. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Op2OfPseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Examples.OpFunctor. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.Associativity. Require Import UniMath.Bicategories.Transformations.Examples.Unitality. Require Import UniMath.Bicategories.Transformations.Examples.Whiskering. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.OtherStructure.DualityInvolution.