Definition poset_enrichment_coprod_is_coprod : is_coprod_enriched E' ys make_poset_enriched_coprod_cocone. Proof. use make_is_coprod_enriched. - intros z P fs. refine (_ · poset_enrichment_coprod_pair _ _ _). simple refine (_ ,, _). + exact (λ x j, pr1 (fs j) x). + abstract (use is_monotone_depfunction_poset_pair ; intro j ; exact (pr2 (fs j))). - abstract (intros z P f g ; use eq_monotone_function ; intros w ; cbn ; apply poset_enrichment_obj_coprod_in_sum). - abstract (intros z P φ₁ φ₂ q ; use eq_monotone_function ; intro w ; use poset_enrichment_coprod_arr_eq ; intro j ; exact (eqtohomot (maponpaths (λ f, pr1 f) (q j)) w)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.CategoryOfPosets. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.PosetEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedInitial. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedBinaryCoproducts. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoproducts. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoequalizers. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCopowers. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Examples.PosetsMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Examples.CartesianMonoidal. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Import MonoidalNotations.