Definition horizontal_dual_inv2cell_weq {x y : B} (v₁ v₂ : x -|-> y) : invertible_2cell (C := 𝕍) v₁ v₂ ≃ invertible_2cell (C := ver_bicat_of_verity_double_bicat horizontal_dual_verity_double_bicat) v₁ v₂. Proof. use weq_iso. - use (λ τ, make_invertible_2cell _). + exact (τ^-1). + use make_is_invertible_2cell. * exact τ. * exact (vcomp_rinv τ). * exact (vcomp_linv τ). - use (λ τ, make_invertible_2cell _). + exact (τ^-1). + use make_is_invertible_2cell. * exact τ. * exact (vcomp_rinv τ). * exact (vcomp_linv τ). - abstract (intro ; apply idpath). - abstract (intro ; apply idpath). 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.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. 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.