Definition is_extensive_one_types : is_extensive one_types_with_biinitial_bincoprod. Proof. intros X Y. split. - refine (_ ,, _ ,, _ ,, _). + apply one_types_isInjective_fully_faithful_1cell ; cbn. apply isweqonpathsincl. apply isinclii1. + apply one_types_isInjective_fully_faithful_1cell ; cbn. apply isweqonpathsincl. apply isinclii2. + exact (one_types_inl_inr X Y). + exact (one_types_inr_inl X Y). - use is_universal_from_pb_alt. + exact one_types_has_pb. + intros Z h. exact (one_types_universal_coprod h). 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.