Definition make_poset_enrichment_coequalizer_is_coequalizer : is_coequalizer_enriched E' f g make_poset_enrichment_coequalizer_cocone. Proof. use make_is_coequalizer_enriched. - abstract (intros w P φ₁ φ₂ q ; use eq_monotone_function ; intro z ; use poset_enrichment_coequalizer_arr_eq ; exact (eqtohomot (maponpaths pr1 q) z)). - intros w P h q. refine (_ · poset_enrichment_coequalizer_to_equalizer EEC f g). simple refine (_ ,, _). + refine (λ z, pr1 h z ,, _). exact (eqtohomot (maponpaths pr1 q) z). + abstract (apply Equalizer_map_monotone ; apply (pr2 h)). - abstract (intros w P h q ; use eq_monotone_function ; intros z ; apply poset_enrichment_obj_from_coequalizer_in). 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.