Proposition structure_to_underlying_binary_prod_map_eq : (λ fg, BinProductArrow C (underlying_BinProduct E' y z (pr2 (EP y z))) (pr1 fg) (pr2 fg)) = (λ fg, structure_to_underlying_binary_prod_map (pr1 fg) (dirprod_pr2 fg)). Proof. use funextsec. intros fg. use is_binary_prod_enriched_arrow_eq. - exact (pr2 (EP y z)). - refine (_ @ !(structure_to_underlying_binary_prod_map_pr1 _ _)). apply (BinProductPr1Commutes C _ _ (underlying_BinProduct E' y z (pr2 (EP y z))) _ _ _). - refine (_ @ !(structure_to_underlying_binary_prod_map_pr2 _ _)). apply (BinProductPr2Commutes C _ _ (underlying_BinProduct 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.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.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. Import MonoidalNotations.