proposition: Proposition coprod_from_underlying_arr_map_eq₂ (f : I_{V} --> PV (λ j, E ⦃ D j , w ⦄)) : coprod_from_underlying_arr_map f · coprod_enriched_to_prod PV a w = f.
proof: Proof. unfold coprod_from_underlying_arr_map. use (ProductArrow_eq _ _ _ (PV (λ j, E ⦃ D j , w ⦄))). unfold coprod_enriched_to_prod. intro j. rewrite !assoc'. etrans. { apply maponpaths. apply (ProductPrCommutes _ _ _ (PV (λ k, E ⦃ D k , w ⦄)) _ _ j). } rewrite enriched_from_arr_precomp. refine (_ @ enriched_from_to_arr E _). apply maponpaths. apply (CoproductInCommutes _ _ _ (make_Coproduct _ _ _ _ _ coprod)). 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