proposition: Proposition is_coprod_enriched_arrow_in {w : C} (f : ∏ (j : J), D j --> w) (j : J) : enriched_coprod_cocone_in a j · is_coprod_enriched_arrow f = f j.
proof: Proof. unfold is_coprod_enriched_arrow, enriched_coprod_cocone_in. use (invmaponpathsweq (make_weq _ (isweq_enriched_from_arr E _ _))) ; cbn. refine (_ @ ProductPrCommutes _ _ _ (is_coprod_enriched_to_Product Ha w) _ (λ j, enriched_from_arr E (f j)) j). cbn. unfold precomp_arr, enriched_coprod_cocone_in. rewrite enriched_from_arr_comp. rewrite !assoc. apply maponpaths_2. rewrite tensor_rinvunitor. rewrite !assoc'. rewrite mon_linvunitor_I_mon_rinvunitor_I. 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 UniMath.CategoryTheory.Limits.Coproducts