Definition cod_disp_prebicat_1_id_comp_cells : disp_prebicat_1_id_comp_cells (prod_bicat B B). Proof. use tpair. - use tpair. + use tpair. * exact (λ X, pr2 X --> pr1 X). * exact (λ X Y fX fY f, invertible_2cell (fX · pr1 f) (pr2 f · fY)). + use tpair. * refine (λ X fX, (runitor _ • linvunitor _ ,, _)). is_iso. * refine (λ X Y Z f g fX fY fZ ff fg, (lassociator _ _ _ • (pr1 ff ▹ _) • rassociator _ _ _ • (_ ◃ pr1 fg) • lassociator _ _ _ ,, _)). cbn ; is_iso. ** exact (pr2 ff). ** exact (pr2 fg). - exact (λ X Y f g α fX fY ff fg, pr1 ff • (pr2 α ▹ _) = (_ ◃ pr1 α) • pr1 fg). 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.