proposition: Proposition enriched_functor_hom_from_arr_eq (x y : C₁) : enriched_from_arr E₂ (τ x) · enriched_functor_left_map_ob EG x y = enriched_from_arr E₂ (τ y) · enriched_functor_right_map_ob EF x y.
proof: Proof. pose (p := Eτ x y). use internal_funext. intros a h. rewrite !tensor_comp_r_id_r. unfold enriched_functor_left_map_ob, enriched_functor_right_map_ob. rewrite !assoc'. rewrite !internal_beta. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. etrans. { apply maponpaths. etrans. { apply maponpaths_2. apply tensor_split'. } rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_split. apply idpath. } etrans. { rewrite !assoc. rewrite <- tensor_sym_mon_braiding. rewrite !assoc'. do 2 apply maponpaths. refine (!(id_left _) @ _). rewrite <- mon_runitor_rinvunitor. rewrite !assoc'. apply maponpaths. rewrite !assoc. exact p. } clear p. rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { apply maponpaths_2. apply tensor_split. } rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite sym_mon_braiding_runitor. rewrite mon_lunitor_linvunitor. rewrite id_left. rewrite <- tensor_split'. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation UniMath.CategoryTheory.EnrichedCats.Examples.HomFunctor UniMath.CategoryTheory.EnrichedCats.Examples.OppositeEnriched UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Products