Lemma linvunitor_natural {X Y : C} {f g : C⟦X, Y⟧} (η : f ==> g) : linvunitor g o η = (η ⋆⋆ id₂ (id₁ X)) o linvunitor f. Proof. use (vcomp_rcancel (lunitor _ )). { apply is_invertible_2cell_lunitor. } rewrite vassocl. rewrite linvunitor_lunitor. use (vcomp_lcancel (lunitor _ )). { apply is_invertible_2cell_lunitor. } repeat rewrite vassocr. rewrite lunitor_linvunitor. rewrite id2_left, id2_right. apply (! lunitor_natural _ _ _ _ _ ). 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.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors.