Definition total_bicat_prod (HB : has_binprod B) (HD₄ : disp_has_binprod HB) : has_binprod (total_bicat D). Proof. intros x y. simple refine (_ ,, _). - exact (total_bicat_prod_cone HB x y HD₄). - use make_binprod_ump. + exact (total_bicat_binprod_ump_1 HB x y HD₄). + exact (total_bicat_binprod_ump_2 HB x y HD₄). + abstract (intros z φ ψ α β ; use subtypePath ; [ intro ; apply HD₁ | ] ; apply binprod_ump_2cell_pr1). + abstract (intros z φ ψ α β ; use subtypePath ; [ intro ; apply HD₁ | ] ; apply binprod_ump_2cell_pr2). + abstract (intros z φ ψ α β γ δ p₁ p₂ q₁ q₂ ; use subtypePath ; [ intro ; apply HD₁ | ] ; exact (binprod_ump_2cell_unique (pr2 (HB (pr1 x) (pr1 y))) (pr1 α) (pr1 β) _ _ (maponpaths pr1 p₁) (maponpaths pr1 p₂) (maponpaths pr1 q₁) (maponpaths pr1 q₂))). 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.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.Examples.EndoMap. 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.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.