proposition: Proposition coprod_from_underlying_arr_map_eq₁ (f : I_{V} --> E ⦃ a , w ⦄) : coprod_from_underlying_arr_map (f · coprod_enriched_to_prod PV a w) = f.
proof: Proof. unfold coprod_from_underlying_arr_map. refine (_ @ enriched_from_to_arr E f). apply maponpaths. use (CoproductArrow_eq _ _ _ (make_Coproduct _ _ _ _ _ coprod)). unfold coprod_enriched_to_prod. intro j. rewrite CoproductInCommutes ; cbn. rewrite (enriched_to_arr_comp E). apply maponpaths. rewrite !assoc'. etrans. { apply maponpaths. apply (ProductPrCommutes _ _ _ (PV (λ k, E ⦃ D k , w ⦄)) _ _ j). } unfold precomp_arr. rewrite !assoc. rewrite tensor_rinvunitor. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite !assoc'. apply maponpaths. rewrite !assoc. 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