Definition duality_involution_inv_mor_inv2cell {x y : B} (f : L x --> L y) : invertible_2cell (#L (duality_involution_inv_mor f)) f. Proof. unfold duality_involution_inv_mor. refine (comp_of_invertible_2cell (inv_of_invertible_2cell (psfunctor_comp L _ _)) _). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (psfunctor_comp L _ _))) _). refine (comp_of_invertible_2cell (rassociator_invertible_2cell _ _ _) _). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (triangle_data_of_duality HL x))) _). refine (comp_of_invertible_2cell (lassociator_invertible_2cell _ _ _) _). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (psnaturality_of η f)) _). refine (comp_of_invertible_2cell (rassociator_invertible_2cell _ _ _) _). refine (comp_of_invertible_2cell _ (runitor_invertible_2cell _)). refine (lwhisker_of_invertible_2cell _ _). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (triangle_data_of_duality HL y)) _). refine (comp_of_invertible_2cell (psfunctor_comp L _ _) _). refine (comp_of_invertible_2cell _ (inv_of_invertible_2cell (psfunctor_id L _))). use (psfunctor_inv2cell L). exact (make_invertible_2cell (to_op2_is_invertible_2cell (invertible_modcomponent_of (unit_unit_inv_of_duality HL) y))). Defined.
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.