Definition make_poset_enrichment_equalizer_is_equalizer : is_equalizer_enriched E' f g make_poset_enrichment_equalizer_cone. Proof. use make_is_equalizer_enriched. - abstract (intros w P φ₁ φ₂ q ; use eq_monotone_function ; intro z ; use poset_enrichment_equalizer_arr_eq ; exact (eqtohomot (maponpaths pr1 q) z)). - intros w P h q. refine (_ · poset_enrichment_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_to_equalizer_pr). 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.Limits.EnrichedTerminal. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedBinaryProducts. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedProducts. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedEqualizers. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedPowers. 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. Import MonoidalNotations.