Proposition enriched_eso_ff_lift_2cell_enrichment : nat_trans_enrichment enriched_eso_ff_lift_2cell_nat_trans (enriched_functor_enrichment L₁) (enriched_functor_enrichment L₂). Proof. use nat_trans_enrichment_via_comp. intros x y. refine (!(id_right _) @ _ @ id_right _). rewrite <- (z_iso_inv_after_z_iso (_ ,, HG (L₁ x) (L₂ y))) ; cbn. rewrite !assoc. apply maponpaths_2. refine (_ @ nat_trans_enrichment_to_comp τ₂ x y @ _) ; cbn. - rewrite !assoc'. apply maponpaths. rewrite <- functor_enrichment_precomp_arr. do 2 apply maponpaths. apply iscontr_enriched_eso_ff_lift_2cell_eq. - rewrite !assoc'. apply maponpaths. rewrite <- functor_enrichment_postcomp_arr. do 2 apply maponpaths. refine (!_). apply iscontr_enriched_eso_ff_lift_2cell_eq. 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.