Proposition is_disp_invertible_2cell_twosided_disp_cat_over_id_inv_laws : twosided_disp_nat_trans_laws (nat_trans_id _) (nat_trans_id _) (pr1 GG) (pr1 FF) is_disp_invertible_2cell_twosided_disp_cat_over_id_inv_data. Proof. intros x₁ x₂ y₁ y₂ f g xy₁ xy₂ fg ; cbn. refine (!_). refine (id_two_disp_right_alt _ @ _). unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite (inv_after_iso_twosided_disp_alt (Hττ x₂ y₂ xy₂)). rewrite two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. etrans. { do 2 apply maponpaths. rewrite assoc_two_disp. apply maponpaths. apply maponpaths_2. apply ττ. } unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp. unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. etrans. { apply maponpaths. do 2 apply maponpaths_2. exact (iso_after_inv_twosided_disp (Hττ x₁ y₁ xy₁)). } unfold transportb_disp_mor2. rewrite !two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite id_two_disp_left. unfold transportb_disp_mor2. rewrite !two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. apply transportf_disp_mor2_idpath. 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.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfDispCats. Require Import UniMath.Bicategories.DisplayedBicats.UnivalenceTechniques. 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.Examples.CatDiag. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor.