Definition one_types_universal_coprod : has_bincoprod_ump (@make_bincoprod_cocone ∁ (hfp_HLevel 3 _ h) (hfp_HLevel 3 _ h) Z (hfpg' inl h) (hfpg' inr h)). Proof. split. - intro W. use make_bincoprod_1cell. + intro z ; cbn. exact (one_types_universal_coprod_map W z (h z) (idpath _)). + use make_invertible_2cell. * intro z. induction z as [ [ x z ] p ] ; cbn in *. exact (!one_types_universal_coprod_map_idpath W z (inl x) p). * apply one_type_2cell_iso. + use make_invertible_2cell. * intro z. induction z as [ [ x z ] p ] ; cbn in *. exact (!one_types_universal_coprod_map_idpath W z (inr x) p). * apply one_type_2cell_iso. - intros W φ ψ α β. use iscontraprop1. + apply one_types_universal_coprod_unique. + simple refine (_ ,, _ ,, _). * exact (one_types_universal_coprod_cell α β). * exact (one_types_universal_coprod_cell_inl α β). * exact (one_types_universal_coprod_cell_inr α β). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.OneTypes. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.Examples.MorphismsInOneTypes. Require Import UniMath.Bicategories.Limits.CommaObjects. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.PullbackFunctions. Require Import UniMath.Bicategories.Limits.Examples.OneTypesLimits. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Colimits.Coproducts. Require Import UniMath.Bicategories.Colimits.Extensive.