Definition duality_involution_preimage_inv2cell : invertible_2cell (duality_transpose_functor x y duality_involution_preimage) f. Proof. unfold duality_involution_preimage ; cbn ; unfold duality_transpose. refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (psfunctor_comp L (η x) (#L f)))) _). refine (comp_of_invertible_2cell (rassociator_invertible_2cell _ _ _) _). refine (comp_of_invertible_2cell (lwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (psnaturality_of ηinv f))) _). refine (comp_of_invertible_2cell (lassociator_invertible_2cell _ _ _) _). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ _) (lunitor_invertible_2cell _)). cbn. refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (triangle_data_of_duality HL x))) _). exact (inv_of_invertible_2cell (invertible_modcomponent_of (unit_unit_inv_of_duality HL) (L x))). 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.