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. 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Subcategory.Core. Require Import UniMath.CategoryTheory.Subcategory.Full. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.ImageEnriched. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DisplayedBicats.Examples.EnrichedCats. Require Import UniMath.Bicategories.OrthogonalFactorization.Orthogonality. Require Import UniMath.Bicategories.OrthogonalFactorization.FactorizationSystem.