Definition trivial_is_disp_invertible_to_is_invertible_2cell {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 gg : c₁ -->[ f] c₂} (β : ff ==> gg) (Hβ : is_disp_invertible_2cell Hα β) : is_invertible_2cell β. Proof. use make_is_invertible_2cell. - exact (pr1 Hβ). - abstract (pose (pr12 Hβ) as p ; cbn in p ; unfold transportb in p ; rewrite transportf_const in p ; exact p). - abstract (pose (pr22 Hβ) as p ; cbn in p ; unfold transportb in p ; rewrite transportf_const in p ; exact p). 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.