Definition id1_fully_faithful {B : bicat} (a : B) : fully_faithful_1cell (id₁ a). Proof. use make_fully_faithful. - apply id1_faithful. - intros z g₁ g₂ αf. simple refine (_ ,, _). + exact (rinvunitor _ • αf • runitor _). + abstract (cbn ; use (vcomp_rcancel (runitor _)) ; [ is_iso | ] ; rewrite !vassocl ; rewrite vcomp_runitor ; rewrite !vassocr ; rewrite runitor_rinvunitor ; rewrite id2_left ; apply idpath). 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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration.