proposition: Proposition coprod_from_underlying_arr_map_eq₁ (f : I_{V} --> E ⦃ a , w ⦄) : coprod_from_underlying_arr_map (f · binary_coprod_enriched_to_coprod BPV a w) = f.
proof: Proof. unfold coprod_from_underlying_arr_map. refine (_ @ enriched_from_to_arr E f). apply maponpaths. use (BinCoproductArrowsEq _ _ _ (make_BinCoproduct C x y a _ _ coprod)). - unfold binary_coprod_enriched_to_coprod. rewrite !assoc'. rewrite !BinProductPr1Commutes. rewrite !BinCoproductIn1Commutes ; cbn. rewrite (enriched_to_arr_comp E). apply maponpaths. unfold precomp_arr. rewrite !assoc. rewrite tensor_rinvunitor. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite enriched_from_to_arr. rewrite <- tensor_split'. apply idpath. - unfold binary_coprod_enriched_to_coprod. rewrite !assoc'. rewrite !BinProductPr2Commutes. rewrite !BinCoproductIn2Commutes ; cbn. rewrite (enriched_to_arr_comp E). apply maponpaths. unfold precomp_arr. rewrite !assoc. rewrite tensor_rinvunitor. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite enriched_from_to_arr. rewrite <- tensor_split'. 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.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts