Definition disp_depprod_bicat_disp_invertible_2cell_weq {a b : B} {f : B ⟦ a, b ⟧} {aa : disp_depprod_bicat a} {bb : disp_depprod_bicat b} (ff : aa -->[ f] bb) (gg : aa -->[ f] bb) : (∏ (i : I), disp_invertible_2cell (id2_invertible_2cell f) (ff i) (gg i)) ≃ disp_invertible_2cell (id2_invertible_2cell f) ff gg. Proof. use make_weq. - exact (disp_depprod_bicat_disp_invertible_2cell_map ff gg). - use isweq_iso. + exact (disp_depprod_bicat_disp_invertible_2cell_inv_map ff gg). + intros x. use funextsec. intro i. use subtypePath. { intro ; apply isaprop_is_disp_invertible_2cell. } apply idpath. + intros x. use subtypePath. { intro ; apply isaprop_is_disp_invertible_2cell. } apply idpath. 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.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence.