Proposition triple_pr2_eq (w₁ x₁ y₁ z₁ : C₁) (w₂ x₂ y₂ z₂ : C₂) : product_enrichment_comp x₁ y₁ z₁ x₂ y₂ z₂ #⊗ identity _ · product_enrichment_pr2_comp w₁ x₁ z₁ w₂ x₂ z₂ = triple_pr2 _ _ _ _ _ _ _ _ · enriched_comp E₂ x₂ y₂ z₂ #⊗ identity _. Proof. use (BinProductArrowsEq _ _ _ (is_cartesian_BinProduct HV _ _)) ; cbn. - rewrite !assoc'. rewrite product_enrichment_pr2_comp_pr1. rewrite !assoc. rewrite cartesian_tensor_pr1. rewrite !assoc'. rewrite product_enrichment_comp_pr2. refine (!_). etrans. { apply maponpaths. apply cartesian_tensor_pr1. } rewrite !assoc. apply maponpaths_2. etrans. { apply (BinProductPr1Commutes _ _ _ (is_cartesian_BinProduct HV _ _)). } use (BinProductArrowsEq _ _ _ (is_cartesian_BinProduct HV _ _)) ; cbn. + etrans. { apply (BinProductPr1Commutes _ _ _ (is_cartesian_BinProduct HV _ _)). } rewrite !assoc'. rewrite product_enrichment_pr2_comp_pr1. apply idpath. + etrans. { apply (BinProductPr2Commutes _ _ _ (is_cartesian_BinProduct HV _ _)). } rewrite !assoc'. rewrite product_enrichment_pr2_comp_pr2. apply idpath. - rewrite !assoc'. rewrite product_enrichment_pr2_comp_pr2. rewrite !assoc. rewrite cartesian_tensor_pr2. rewrite !assoc'. rewrite id_left. refine (!_). etrans. { apply maponpaths. apply cartesian_tensor_pr2. } rewrite id_right. etrans. { apply (BinProductPr2Commutes _ _ _ (is_cartesian_BinProduct HV _ _)). } apply idpath. 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.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Import MonoidalNotations.