Proposition local_equivalence_twocell_linvunitor {x y : B₁} (f : F x --> F y) : linvunitor (local_equivalence_inv_onecell f) = local_equivalence_inv_twocell (linvunitor f) • local_equivalence_inv_onecell_comp _ _ • (local_equivalence_inv_onecell_id x ▹ _). Proof. refine (!(id2_left _) @ _ @ id2_right _). rewrite <- lunitor_linvunitor. rewrite !vassocr. apply maponpaths_2. rewrite local_equivalence_twocell_lunitor. rewrite !vassocl. refine (!_). etrans. { do 2 apply maponpaths. rewrite !vassocr. rewrite rwhisker_vcomp. rewrite vcomp_rinv. rewrite id2_rwhisker. rewrite id2_left. apply idpath. } etrans. { apply maponpaths. rewrite !vassocr. rewrite vcomp_rinv. apply id2_left. } rewrite <- local_equivalence_inv_twocell_comp. rewrite linvunitor_lunitor. rewrite local_equivalence_inv_twocell_id. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties.