Proposition structure_to_underlying_binary_coprod_map_eq : (λ fg, BinCoproductArrow (underlying_BinCoproduct E' y z (pr2 (EP y z))) (pr1 fg) (pr2 fg)) = (λ fg, structure_to_underlying_binary_coprod_map (pr1 fg) (dirprod_pr2 fg)). Proof. use funextsec. intros fg. use is_binary_coprod_enriched_arrow_eq. - exact (pr2 (EP y z)). - refine (_ @ !(structure_to_underlying_binary_coprod_map_in1 _ _)). apply (BinCoproductIn1Commutes C _ _ (underlying_BinCoproduct E' y z (pr2 (EP y z))) _ _ _). - refine (_ @ !(structure_to_underlying_binary_coprod_map_in2 _ _)). apply (BinCoproductIn2Commutes C _ _ (underlying_BinCoproduct E' y z (pr2 (EP y z))) _ _ _). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.StructureEnriched. 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.StructuresMonoidal. 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.