Proposition square_between_companions_unit : lunitor (id_h y) ▿s (linvunitor h ▵s γ₁ ⋆h unit_are_companions c₂) = unit_are_companions c₁. Proof. unfold γ₁, square_between_companions. etrans. { do 3 apply maponpaths. refine (!_). apply runitor_v_comp_square''. } rewrite <- lwhisker_hcomp_square. rewrite <- rwhisker_hcomp_square. etrans. { do 4 apply maponpaths. rewrite lrwhisker_hcomp_square. rewrite <- rwhisker_square_comp. rewrite runitor_rinvunitor. rewrite rwhisker_square_id. rewrite double_bicat_interchange. apply maponpaths_2. apply are_companions_right'. } etrans. { apply maponpaths. rewrite dwhisker_uwhisker_square. rewrite <- uwhisker_vcomp_square. rewrite rwhisker_uwhisker_square. rewrite lwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite linvunitor_lunitor. rewrite uwhisker_square_id. apply idpath. } rewrite <- ud_whisker_vcomp_square. rewrite lunitor_v_comp_square'. rewrite <- rwhisker_square_comp. rewrite runitor_lunitor_identity. rewrite linvunitor_lunitor. rewrite rwhisker_square_id. rewrite <- lwhisker_square_comp. rewrite linvunitor_lunitor. rewrite lwhisker_square_id. rewrite <- id_h_square_bicat_id. rewrite runitor_h_comp_square'. rewrite !dwhisker_uwhisker_square. rewrite <- dwhisker_square_comp. rewrite <- uwhisker_square_comp. rewrite lunitor_runitor_identity. rewrite !rinvunitor_runitor. rewrite dwhisker_square_id. rewrite uwhisker_square_id. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. 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.DoubleBicat.CompanionPairs.