Definition has_pb_univ_cat_with_initial : has_pb univ_cat_with_initial. Proof. use subbicat_has_pb. - exact has_pb_bicat_of_univ_cats. - exact univalent_cat_is_univalent_2. - intros C₁ C₂ C₃ F G. apply initial_category_iso_comma. + exact (pr22 F). + exact (pr22 G). + exact (pr12 C₁). + exact (pr12 C₂). - intros C₁ C₂ C₃ F G. apply iso_comma_pr1_preserves_initial. + exact (pr22 F). + exact (pr22 G). + exact (pr12 C₁). + exact (pr12 C₂). - intros C₁ C₂ C₃ F G. apply iso_comma_pr2_preserves_initial. + exact (pr22 F). + exact (pr22 G). + exact (pr12 C₁). + exact (pr12 C₂). - intros C₁ C₂ C₃ F G q. apply iso_comma_ump1_preserves_initial. + exact (pr22 F). + exact (pr22 (pb_cone_pr1 q)). + exact (pr22 (pb_cone_pr2 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.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Categories.EilenbergMoore. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.Examples.UnitCategoryLimits. Require Import UniMath.CategoryTheory.Limits.Examples.CategoryProductLimits. Require Import UniMath.CategoryTheory.Limits.Examples.IsoCommaLimits. Require Import UniMath.CategoryTheory.Limits.Examples.EilenbergMooreLimits. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. 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.BicatOfUnivCatsLimits. Require Import UniMath.Bicategories.Limits.Examples.TotalBicategoryLimits. Require Import UniMath.Bicategories.Limits.Examples.DispConstructionsLimits. Require Import UniMath.Bicategories.Limits.Examples.SubbicatLimits. 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.