Proposition isaprop_enriched_eso_ff_lift_ob : isaprop (∑ (y : C₁), z_iso (G y) (Φ₂ x)). Proof. use invproofirrelevance. intros φ₁ φ₂. use total2_paths_f. - apply (isotoid _). + apply univalent_category_is_univalent. + exact (make_z_iso' _ (fully_faithful_reflects_iso_proof _ _ _ G_fully_faithful _ _ (z_iso_comp (pr2 φ₁) (z_iso_inv_from_z_iso (pr2 φ₂))))). - use subtypePath. { intro. apply isaprop_is_z_isomorphism. } etrans. { apply transportf_z_iso_functors. } rewrite functor_on_inv_from_z_iso. use z_iso_inv_on_right. refine (!_). etrans. { apply maponpaths_2. cbn. rewrite idtoiso_isotoid. apply (homotweqinvweq (make_weq _ (G_fully_faithful _ _)) _). } rewrite !assoc'. rewrite z_iso_after_z_iso_inv. apply id_right. 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.