Proposition eq_companion_of_hor {B : verity_double_bicat} (H : vertically_saturated B) (HB_2_1 : locally_univalent_verity_double_bicat B) {x y : B} {h : x --> y} {φ₁ φ₂ : hor_companion h} (p : pr1 φ₁ = pr1 φ₂) (q₁ : lunitor _ ▿s (linvunitor _ ▵s (v_sq_idtoiso_2_1 p ⋆h unit_are_companions φ₂)) = unit_are_companions φ₁) (q₂ : runitor _ ▿s (linvunitor _ ▵s (counit_are_companions φ₁ ⋆h v_sq_idtoiso_2_1 p)) = counit_are_companions φ₂) : φ₁ = φ₂. Proof. induction φ₁ as [ v₁ [ c₁ [ Hh₁ Hv₁ ] ] ]. induction φ₂ as [ v₂ [ c₂ [ Hh₂ Hv₂ ] ] ]. cbn in p. induction p. cbn in *. apply maponpaths. use eq_are_companions. - refine (!q₁ @ _). unfold vertical_cell_to_square. rewrite lwhisker_square_id. rewrite lunitor_h_comp_square'. rewrite !dwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite <- dwhisker_square_comp. rewrite !linvunitor_lunitor. rewrite uwhisker_square_id, dwhisker_square_id. apply idpath. - refine (_ @ q₂). unfold vertical_cell_to_square. rewrite lwhisker_square_id. rewrite runitor_h_comp_square'. rewrite !dwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite <- dwhisker_square_comp. rewrite runitor_lunitor_identity. rewrite linvunitor_lunitor. rewrite rinvunitor_runitor. rewrite uwhisker_square_id, dwhisker_square_id. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.UnderlyingCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.LocalUnivalence. Require Import UniMath.Bicategories.DoubleCategories.Examples.TransposeDoubleBicat.