Proposition poset_to_underlying_binary_coprod_map_eq (f : x₁ --> y) (g : x₂ --> y) : BinCoproductArrow (underlying_BinCoproduct E' x₁ x₂ (pr2 (EP x₁ x₂))) f g = poset_to_underlying_binary_coprod_map f g. Proof. use is_binary_coprod_enriched_arrow_eq. - exact (pr2 (EP x₁ x₂)). - refine (_ @ !(poset_to_underlying_binary_coprod_map_in1 f g)). apply (BinCoproductIn1Commutes C _ _ (underlying_BinCoproduct E' x₁ x₂ (pr2 (EP x₁ x₂))) _ f g). - refine (_ @ !(poset_to_underlying_binary_coprod_map_in2 f g)). apply (BinCoproductIn2Commutes C _ _ (underlying_BinCoproduct E' x₁ x₂ (pr2 (EP x₁ x₂))) _ f g). 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.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.