Proposition enriched_eso_ff_lift_comm2_nat_trans_enrichment : nat_trans_enrichment enriched_eso_ff_lift_comm2_nat_trans (enriched_functor_enrichment (enriched_eso_ff_lift_enriched_functor · G)) (enriched_functor_enrichment Φ₂). Proof. use nat_trans_enrichment_via_comp. intros x y ; cbn. refine (!(id_right _) @ _). rewrite !assoc'. do 2 apply maponpaths. refine (!_). etrans. { apply maponpaths. rewrite !assoc. rewrite z_iso_after_z_iso_inv. apply id_left. } rewrite <- postcomp_arr_comp. rewrite <- postcomp_arr_id. apply maponpaths. apply z_iso_after_z_iso_inv. 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.