Definition subbicat_disp_has_pb (HB : has_pb B) (Hcone : ∏ (x y z : subbicat P₀ P₁ Pid Pcomp) (f : x --> z) (g : y --> z), P₀ (pr1 (HB _ _ _ (pr1 f) (pr1 g)))) (Hπ₁ : ∏ (x y z : subbicat P₀ P₁ Pid Pcomp) (f : x --> z) (g : y --> z), P₁ _ _ (Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (HB _ _ _ (pr1 f) (pr1 g))))) (Hπ₂ : ∏ (x y z : subbicat P₀ P₁ Pid Pcomp) (f : x --> z) (g : y --> z), P₁ _ _ (Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (HB _ _ _ (pr1 f) (pr1 g))))) (H_ump_mor : ∏ (x y z : subbicat P₀ P₁ Pid Pcomp) (f : x --> z) (g : y --> z) (q : pb_cone f g), P₁ _ _ (pr121 q) (Hcone x y z f g) (pb_ump_mor (pr2 (HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone _ q))) : disp_has_pb (disp_subbicat P₀ P₁ Pid Pcomp) HB. Proof. intros x y z f g. simple refine (_ ,, _ ,, _ ,, _). - exact (Hcone x y z f g ,, tt). - exact (tt ,, Hπ₁ x y z f g). - exact (tt ,, Hπ₂ x y z f g). - exact (λ q, tt ,, H_ump_mor x y z f g q). 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.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.MonadsLax. Require Import UniMath.Bicategories.Limits.Final. Require Import UniMath.Bicategories.Limits.Products. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.EilenbergMooreObjects. Require Import UniMath.Bicategories.Limits.Examples.TotalBicategoryLimits. Require Import UniMath.Bicategories.Monads.Examples.MonadsInTotalBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.MonadInclusion.