proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfDispCats UniMath.Bicategories.DisplayedBicats.UnivalenceTechniques UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Examples.CatDiag UniMath.Bicategories.DisplayedBicats.DispPseudofunctor