Definition enriched_cat_fully_faithful_to_fully_faithful_1cell {E₁ E₂ : enriched_cat V} (F : enriched_functor E₁ E₂) (HF : fully_faithful_enriched_functor (enriched_functor_enrichment F)) : fully_faithful_1cell F. Proof. use make_fully_faithful. - apply enriched_cat_faithful_is_faithful_1cell. exact (fully_faithful_enriched_functor_to_faithful _ HF). - intros E₀ G₁ G₂ β. simple refine (_ ,, _). + exact (enriched_cat_fully_faithful_to_fully_faithful_1cell_cell HF β). + exact (enriched_cat_fully_faithful_to_fully_faithful_1cell_eq HF β). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentAdjunction. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.UnitEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FullSubEnriched. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.Examples.EnrichedCats. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.Examples.MorphismsInBicatOfUnivCats. Import MonoidalNotations.