Definition disp_bicat_of_univ_disp_cats_is_disp_invertible_2cell {C C' : bicat_of_univ_cats} {F : C --> C'} {D : disp_bicat_of_univ_disp_cats C} {D' : disp_bicat_of_univ_disp_cats C'} {FF : D -->[ F ] D'} {GG : D -->[ F ] D'} (αα : FF ==>[ id₂ F ] GG) (Hαα : is_disp_nat_z_iso (nat_z_iso_id _) αα) : is_disp_invertible_2cell (id2_invertible_2cell F) αα. Proof. use tpair. - exact (pointwise_inverse_disp_nat_trans αα Hαα). - split. + abstract (cbn ; simpl in * ; use (@disp_nat_trans_eq C C') ; intros x xx ; cbn ; refine (inv_mor_after_z_iso_disp (Hαα x xx) @ _) ; refine (!_) ; refine (@disp_nat_trans_transportf _ _ _ _ _ _ _ _ (!(@id2_left bicat_of_univ_cats _ _ _ _ (nat_trans_id F))) _ _ _ _ _ @ _) ; apply transportf_paths ; apply homset_property). + abstract (cbn ; simpl in * ; use (@disp_nat_trans_eq C C') ; intros x xx ; cbn ; refine (z_iso_disp_after_inv_mor (Hαα x xx) @ _) ; refine (!_) ; refine (@disp_nat_trans_transportf _ _ _ _ _ _ _ _ (!(@id2_left bicat_of_univ_cats _ _ _ _ (nat_trans_id F))) _ _ _ _ _ @ _) ; apply transportf_paths ; apply homset_property). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. 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.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Adjunctions. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.Adjunctions. 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.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.