proposition: Proposition is_prod_enriched_arrow_pr {w : C} (f : ∏ (j : J), w --> D j) (j : J) : is_prod_enriched_arrow f · enriched_prod_cone_pr a j = f j.
proof: Proof. unfold is_prod_enriched_arrow, enriched_prod_cone_pr. use (invmaponpathsweq (make_weq _ (isweq_enriched_from_arr E _ _))) ; cbn. refine (_ @ ProductPrCommutes _ _ _ (is_prod_enriched_to_Product Ha w) _ (λ j, enriched_from_arr E (f j)) j). cbn. unfold postcomp_arr, enriched_prod_cone_pr. rewrite enriched_from_arr_comp. rewrite !assoc. apply maponpaths_2. rewrite tensor_linvunitor. rewrite !assoc'. apply maponpaths. rewrite <- tensor_split. rewrite !enriched_from_to_arr. 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.Monoidal.Categories UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.Limits.Products