proposition: Proposition prod_from_underlying_arr_map_eq₂ (f : I_{V} --> PV (λ j, E ⦃ w , D j ⦄)) : prod_from_underlying_arr_map f · prod_enriched_to_prod PV a w = f.
proof: Proof. unfold prod_from_underlying_arr_map. use (ProductArrow_eq _ _ _ (PV (λ j, E ⦃ w, D j ⦄))). unfold prod_enriched_to_prod. intro j. rewrite !assoc'. etrans. { apply maponpaths. apply (ProductPrCommutes _ _ _ (PV (λ k, E ⦃ w , D k ⦄)) _ _ j). } rewrite enriched_from_arr_postcomp. refine (_ @ enriched_from_to_arr E _). apply maponpaths. apply (ProductPrCommutes _ _ _ (make_Product _ _ _ _ _ prod)). 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