Definition disp_bicat_of_cleaving_disp_invertible_2cell_pointwise_inv {C C' : bicat_of_univ_cats} {F G : C --> C'} {α : F ==> G} (Hα : is_invertible_2cell α) {D : disp_bicat_of_cleaving C} {D' : disp_bicat_of_cleaving C'} {FF : D -->[ F ] D'} {GG : D -->[ G ] D'} (αα : FF ==>[ α ] GG) (Hαα : is_disp_invertible_2cell Hα αα) {x : (C : univalent_category)} (xx : (pr1 D : disp_univalent_category _) x) : is_z_iso_disp (make_z_iso' (pr1 α x) (is_invertible_2cell_to_is_nat_z_iso _ Hα x)) (pr11 αα x xx). Proof. simple refine (_ ,, _). - exact (pr111 Hαα x xx). - split. + abstract (unfold transportb ; etrans ; [ apply (maponpaths (λ z, pr11 z x xx) (pr22 Hαα)) |] ; unfold transportb ; etrans ; [ refine (maponpaths (λ z, pr1 z x xx) _) ; exact (pr1_transportf (!(vcomp_linv Hα)) (disp_nat_trans_id (pr11 GG),, tt)) | ]; etrans ; [ exact (@disp_nat_trans_transportf _ _ _ _ _ _ _ _ (!(vcomp_linv Hα)) _ _ (disp_nat_trans_id (pr11 GG)) x xx) | ] ; apply maponpaths_2 ; apply homset_property). + abstract (unfold transportb ; etrans ; [ apply (maponpaths (λ z, pr11 z x xx) (pr12 Hαα)) |] ; unfold transportb ; etrans ; [ refine (maponpaths (λ z, pr1 z x xx) _) ; exact (pr1_transportf (!(vcomp_rinv Hα)) (disp_nat_trans_id (pr11 FF),, tt)) | ] ; etrans ; [ exact (@disp_nat_trans_transportf _ _ _ _ _ _ _ _ (!(vcomp_rinv Hα)) _ _ (disp_nat_trans_id (pr11 FF)) x xx) | ] ; apply maponpaths_2 ; 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.