Definition cod_disp_invertible_invertible_2_cell {c₁ c₂ : B} {f : B ⟦ c₁, c₂ ⟧} {φ₁ : cod_disp_bicat B c₁} {φ₂ : cod_disp_bicat B c₂} {ψ₁ ψ₂ : φ₁ -->[ f] φ₂} : disp_invertible_2cell (id2_invertible_2cell f) ψ₁ ψ₂ → ∑ (α : invertible_2cell (pr1 ψ₁) (pr1 ψ₂)), coherent_homot (id2_invertible_2cell f) (pr1 α). Proof. intro α. simple refine ((_ ,, _) ,, _). - exact (pr11 α). - simpl. use make_is_invertible_2cell. + exact (pr1 (disp_inv_cell α)). + abstract (assert (p := maponpaths pr1 (disp_vcomp_rinv α)); cbn in p ; (etrans; [exact p |]) ; clear p ; unfold transportb ; (etrans ; [exact (pr1_transportf (! id2_left (id₂ f)) (id₂ (pr1 ψ₁),, disp_id2 (pr2 ψ₁))) |]) ; cbn ; rewrite transportf_const ; apply idpath). + abstract (assert (p := maponpaths pr1 (disp_vcomp_linv α)); cbn in p ; (etrans; [exact p |]) ; clear p ; unfold transportb ; (etrans ; [exact (pr1_transportf (! id2_left (id₂ f)) (id₂ (pr1 ψ₂),, disp_id2 (pr2 ψ₂))) |]) ; cbn ; rewrite transportf_const ; apply idpath). - exact (pr21 α). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. 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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.OneTypes. 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. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Trivial. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma.