Lemma representable_faithful {C : bicat} {X Y Z : C} {f : C⟦X,Y⟧} {g : C⟦Y,X⟧} (η : id₁ X ==> g ∘ f) (k₁ k₂ : C⟦Z,X⟧) (α β : k₁ ==> k₂) (Hη : is_invertible_2cell η) : f ◅ α = f ◅ β -> α = β. Proof. intros Hαβ. rewrite (whisker_l_iso_id₁ η _ _ α Hη). rewrite (whisker_l_iso_id₁ η _ _ β Hη). do 2 apply maponpaths. apply (maponpaths (fun z => _ o z)). apply (whisker_l_eq k₁ k₂ α β Hαβ). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence.