Proposition to_disp_left_adjequiv_hor_comp_inv_laws : double_functor_hor_comp_laws to_disp_left_adjequiv_hor_comp_inv_data. Proof. intros x₁ x₂ y₁ y₂ z₁ z₂ v₁ v₂ v₃ h₁ h₂ k₁ k₂ s₁ s₂ ; cbn. pose (pr2 τ x₁ x₂ y₁ y₂ z₁ z₂ v₁ v₂ v₃ h₁ h₂ k₁ k₂ s₁ s₂) 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 _ _ _ _ _)). } 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. apply (pr2 τ). } 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.
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.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.Examples.DispBicatOfTwoSidedDispCat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleTransformation. Section FromAdjEquivHorId. End FromAdjEquivHorId. Section FromAdjEquivHorComp. End FromAdjEquivHorComp.