Definition structure_enrichment_prod_is_prod : is_prod_enriched E' ys make_structure_enriched_prod_cone. Proof. use make_is_prod_enriched. - intros z X fs. refine (_ · structure_enrichment_prod_pair _ _ _). simple refine (_ ,, _). + exact (λ x j, pr1 (fs j) x). + abstract (use hset_struct_type_prod_pair ; intro j ; exact (pr2 (fs j))). - abstract (intros z X f g ; use eq_mor_hset_struct ; intros w ; cbn ; apply structure_enrichment_obj_prod_pair_pr). - abstract (intros z X φ₁ φ₂ q ; use eq_mor_hset_struct ; intro w ; use structure_enrichment_prod_arr_eq ; intro j ; exact (eqtohomot (maponpaths (λ f, pr1 f) (q j)) w)). Defined.
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.