Definition trivial_is_invertible_2cell_to_is_disp_invertible {B C : bicat} {b₁ b₂ : B} {f g : B ⟦ b₁, b₂ ⟧} {α : f ==> g} (Hα : is_invertible_2cell α) {c₁ : trivial_displayed_bicat B C b₁} {c₂ : trivial_displayed_bicat B C b₂} {ff : c₁ -->[ f ] c₂} {gg : c₁ -->[ g ] c₂} (β : ff ==> gg) (Hβ : is_invertible_2cell β) : is_disp_invertible_2cell Hα β. Proof. simple refine (_ ,, (_ ,, _)). - exact (Hβ^-1). - abstract (unfold transportb ; cbn ; rewrite transportf_const ; apply vcomp_rinv). - abstract (unfold transportb ; cbn ; rewrite transportf_const ; apply vcomp_linv). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.