proposition: Proposition coprod_from_underlying_arr_map_eq₂ (f : I_{V} --> BPV (E ⦃ x , w ⦄) (E ⦃ y , w ⦄)) : coprod_from_underlying_arr_map f · binary_coprod_enriched_to_coprod BPV a w = f.
proof: Proof. unfold coprod_from_underlying_arr_map. use (BinProductArrowsEq _ _ _ (BPV (E ⦃ x , w ⦄) (E ⦃ y , w ⦄))). - unfold binary_coprod_enriched_to_coprod. rewrite !assoc'. rewrite !BinProductPr1Commutes. rewrite enriched_from_arr_precomp. refine (_ @ enriched_from_to_arr E _). apply maponpaths. apply (BinCoproductIn1Commutes _ _ _ (make_BinCoproduct C x y a _ _ coprod)). - unfold binary_coprod_enriched_to_coprod. rewrite !assoc'. rewrite !BinProductPr2Commutes. rewrite enriched_from_arr_precomp. refine (_ @ enriched_from_to_arr E _). apply maponpaths. apply (BinCoproductIn2Commutes _ _ _ (make_BinCoproduct C x y a _ _ 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.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts