Definition fully_faithful_1cell {B : bicat} {a b : B} (f : a --> b) : UU := (∏ (z : B) (g₁ g₂ : z --> a) (α₁ α₂ : g₁ ==> g₂), α₁ ▹ f = α₂ ▹ f → α₁ = α₂) × (∏ (z : B) (g₁ g₂ : z --> a) (αf : g₁ · f ==> g₂ · f), ∑ (α : g₁ ==> g₂), α ▹ f = αf).
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.