Definition discrete_1cell_weq_faithful_locally_groupoid {B : bicat} (HB : locally_groupoid B) {a b : B} (f : a --> b) : discrete_1cell f ≃ faithful_1cell f. Proof. use weqimplimpl. - exact (λ Hf, pr1 Hf). - intro Hf. split. + exact Hf. + exact (conservative_1cell_locally_groupoid HB f). - apply isaprop_discrete_1cell. - apply isaprop_faithful_1cell. 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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.FullyFaithful.