Definition representable_id_inv2cell (X : C) : invertible_modification_data (id_pstrans (representable C_is_univalent_2_1 X)) (representable1 C_is_univalent_2_1 (id₁ X)). Proof. intro Y. use make_invertible_2cell. - use make_nat_trans. * intro f. cbn in *. apply rinvunitor. * abstract (intros f g η ; cbn in * ; rewrite rwhisker_hcomp ; apply rinvunitor_natural). - use make_is_invertible_2cell. + use make_nat_trans. * intro f. cbn in *. apply runitor. * abstract (intros f g η ; cbn ; apply vcomp_runitor). + use nat_trans_eq. { exact (pr2 C Y X). } abstract (intro f ; cbn in * ; apply rinvunitor_runitor). + use nat_trans_eq. { exact (pr2 C Y X). } abstract (intro f ; cbn in * ; apply runitor_rinvunitor). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.PseudoFunctors.Display.Base. Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.Identitor. Require Import UniMath.Bicategories.PseudoFunctors.Display.Compositor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.PseudoFunctors.Representable. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification.