proposition: Proposition to_disp_left_adjequiv_hor_id_inv_laws : double_functor_hor_id_laws to_disp_left_adjequiv_hor_id_inv_data.
proof: Proof. intros x y f ; cbn. pose (pr2 τ x y f) as p ; cbn in p. refine (!_). etrans. { apply maponpaths. apply id_two_disp_right_alt. } unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. etrans. { do 3 apply maponpaths. exact (inv_after_iso_twosided_disp_alt (Hτ y)). } rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. etrans. { do 2 apply maponpaths. rewrite assoc_two_disp. apply maponpaths. apply maponpaths_2. exact (pr2 τ x y f). } unfold transportb_disp_mor2. rewrite !two_disp_pre_whisker_f. rewrite !two_disp_post_whisker_f. 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. apply Hτ. } 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.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.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.Examples.DispBicatOfTwoSidedDispCat UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat UniMath.Bicategories.DisplayedBicats.Examples.FullSub UniMath.Bicategories.DisplayedBicats.Examples.Prod UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor UniMath.Bicategories.DoubleCategories.Core.DoubleTransformation