Proposition locally_groupoid_duality_involution_triangle_law {x y : B} (f : x --> y) : (id₂ (id₁ x) ▹ f) • id₂ (id₁ x · f) = (((lunitor f • rinvunitor f) • (f ◃ id₂ (id₁ y))) • id₂ (f · id₁ y)) • (inv_B x y (id₁ x · f) (f · id₁ y) (lunitor f • rinvunitor f))^-1. Proof. rewrite id2_rwhisker, lwhisker_id2. rewrite !id2_left, !id2_right. use vcomp_move_L_pM ; [ is_iso | ]. cbn. rewrite id2_right. refine (_ @ id2_right _). use vcomp_move_L_pM ; [ is_iso | ]. cbn. rewrite !vassocl. etrans. { apply maponpaths. rewrite !vassocr. rewrite rinvunitor_runitor. apply id2_left. } apply lunitor_linvunitor. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. 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.Examples.OneTypes. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Limits.Products. Require Import UniMath.Bicategories.Limits.Examples.OneTypesLimits. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. 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.ConstProduct. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.OtherStructure.DualityInvolution. Require Import UniMath.Bicategories.OtherStructure.Exponentials.