Definition inv2cell_are_companions {B : verity_double_bicat} (H : vertically_saturated B) {x y : B} {h₁ h₂ : x --> y} (τ : invertible_2cell h₁ h₂) {v₁ v₂ : x -|-> y} (c₁ : are_companions h₁ v₁) (c₂ : are_companions h₂ v₂) : invertible_2cell v₁ v₂. Proof. use make_invertible_2cell. - exact (cell_are_companions H (τ^-1) c₁ c₂). - use make_is_invertible_2cell. + exact (cell_are_companions H τ c₂ c₁). + abstract (rewrite <- cell_are_companions_comp ; rewrite vcomp_rinv ; apply cell_are_companions_id). + abstract (rewrite <- cell_are_companions_comp ; rewrite vcomp_linv ; apply cell_are_companions_id). Defined.
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.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. 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.CompanionPairs.