Proposition poset_to_underlying_binary_prod_map_eq (f : x --> y₁) (g : x --> y₂) : BinProductArrow C (underlying_BinProduct E' y₁ y₂ (pr2 (EP y₁ y₂))) f g = poset_to_underlying_binary_prod_map f g. Proof. use is_binary_prod_enriched_arrow_eq. - exact (pr2 (EP y₁ y₂)). - refine (_ @ !(poset_to_underlying_binary_prod_map_pr1 f g)). apply (BinProductPr1Commutes C _ _ (underlying_BinProduct E' y₁ y₂ (pr2 (EP y₁ y₂))) _ f g). - refine (_ @ !(poset_to_underlying_binary_prod_map_pr2 f g)). apply (BinProductPr2Commutes C _ _ (underlying_BinProduct E' y₁ y₂ (pr2 (EP y₁ y₂))) _ 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.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.