proposition: Proposition is_prod_enriched_arrow_eq {w : C} {f g : w --> a} (q : ∏ (j : J), f · enriched_prod_cone_pr a j = g · enriched_prod_cone_pr a j) : f = g.
proof: Proof. refine (!(enriched_to_from_arr E _) @ _ @ enriched_to_from_arr E _). apply maponpaths. use (ProductArrow_eq _ _ _ (is_prod_enriched_to_Product Ha w)). intro j. cbn. unfold postcomp_arr. rewrite !assoc. rewrite !tensor_linvunitor. rewrite !assoc'. rewrite !(maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite <- !tensor_split. use (invmaponpathsweq (make_weq _ (isweq_enriched_to_arr E _ _))) ; cbn. rewrite !assoc. rewrite <- !(enriched_to_arr_comp E). exact (q j). 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