proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Core UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts