proposition: Proposition enriched_eso_ff_lift_comm1_nat_trans_enrichment : nat_trans_enrichment enriched_eso_ff_lift_comm1_nat_trans (enriched_functor_enrichment (F · enriched_eso_ff_lift_enriched_functor)) (enriched_functor_enrichment Φ₁).
proof: Proof. use nat_trans_enrichment_via_comp. intros x y ; cbn. rewrite !assoc'. refine (!(id_right _) @ _ @ id_right _). rewrite <- !(z_iso_inv_after_z_iso (fully_faithful_enriched_functor_z_iso HG _ _)). rewrite !assoc. apply maponpaths_2 ; cbn. rewrite !assoc'. etrans. { apply maponpaths. rewrite <- functor_enrichment_precomp_arr. do 2 apply maponpaths. exact (enriched_eso_ff_lift_comm1_nat_trans_data_eq x). } refine (!_). etrans. { do 5 apply maponpaths. rewrite <- functor_enrichment_postcomp_arr. do 2 apply maponpaths. exact (enriched_eso_ff_lift_comm1_nat_trans_data_eq y). } etrans. { do 4 apply maponpaths. rewrite !assoc. rewrite z_iso_after_z_iso_inv. apply id_left. } rewrite <- postcomp_arr_comp. etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite z_iso_after_z_iso_inv. rewrite id_left. apply idpath. } rewrite precomp_postcomp_arr. etrans. { rewrite !assoc. apply maponpaths_2. exact (!(nat_trans_enrichment_to_comp (τ^-1 : enriched_nat_trans _ _) x y)). } cbn. rewrite !assoc'. do 2 apply maponpaths. rewrite <- precomp_arr_comp. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Subcategory.Core UniMath.CategoryTheory.Subcategory.Full UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation UniMath.CategoryTheory.EnrichedCats.Examples.ImageEnriched UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.DisplayedBicats.Examples.EnrichedCats UniMath.Bicategories.OrthogonalFactorization.Orthogonality UniMath.Bicategories.OrthogonalFactorization.FactorizationSystem