proposition: Proposition enriched_functor_hom_id_eq (x y : C₁) : enriched_id E₂ (F x) · enriched_functor_left_map_ob EF x y = enriched_id E₂ (F y) · enriched_functor_right_map_ob EF x y.
proof: Proof. 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. rewrite !assoc. rewrite <- tensor_comp_mor. rewrite id_right. rewrite tensor_split'. rewrite !assoc'. rewrite <- enrichment_id_right. apply idpath. } refine (!_). etrans. { rewrite !assoc. rewrite <- tensor_comp_mor. rewrite id_right. rewrite tensor_split. rewrite !assoc'. rewrite <- enrichment_id_left. apply idpath. } rewrite tensor_lunitor. rewrite tensor_runitor. rewrite !assoc. do 2 apply maponpaths_2. rewrite sym_mon_braiding_runitor. 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