Proposition enriched_eso_ff_lift_comm1_nat_trans_laws : is_nat_trans _ _ enriched_eso_ff_lift_comm1_nat_trans_data. Proof. intros x y f. use (invmaponpathsweq (make_weq _ (G_fully_faithful _ _))) ; cbn. rewrite !functor_comp. etrans. { apply maponpaths. apply enriched_eso_ff_lift_comm1_nat_trans_data_eq. } refine (!_). etrans. { apply maponpaths_2. apply enriched_eso_ff_lift_comm1_nat_trans_data_eq. } etrans. { rewrite !assoc'. apply maponpaths. exact (!(nat_trans_ax (nat_z_iso_inv τ_iso) _ _ f)). } refine (!_). rewrite !assoc. apply maponpaths_2. cbn. etrans. { apply maponpaths_2. exact (enriched_eso_ff_lift_mor_eq _). } refine (_ @ id_right _). rewrite !assoc'. do 2 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.