Proposition make_poset_enrichment_laws : poset_enrichment_laws make_poset_enrichment_data. Proof. repeat split. - intros x y z f₁ f₂ g p ; cbn. rewrite !enriched_from_arr_comp ; cbn. pose (Ef₁ := enriched_from_arr E f₁ : monotone_function _ _). pose (Ef₂ := enriched_from_arr E f₂ : monotone_function _ _). pose (Eg := enriched_from_arr E g : monotone_function _ _). use (pr2 (enriched_comp E x y z) (Eg tt ,, Ef₁ tt) (Eg tt ,, Ef₂ tt)). split. + apply refl_PartialOrder. + exact p. - intros x y z f g₁ g₂ p ;cbn. rewrite !enriched_from_arr_comp ; cbn. pose (Ef := enriched_from_arr E f : monotone_function _ _). pose (Eg₁ := enriched_from_arr E g₁ : monotone_function _ _). pose (Eg₂ := enriched_from_arr E g₂ : monotone_function _ _). use (pr2 (enriched_comp E x y z) (Eg₁ tt ,, Ef tt) (Eg₂ tt ,, Ef tt)). split. + exact p. + apply refl_PartialOrder. Qed.
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.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.CategoryOfPosets. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Examples.PosetsMonoidal. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Section FromPosetEnrichment. End FromPosetEnrichment.