Proposition local_equivalence_twocell_rinvunitor {x y : B₁} (f : F x --> F y) : rinvunitor (local_equivalence_inv_onecell f) = local_equivalence_inv_twocell (rinvunitor f) • local_equivalence_inv_onecell_comp _ _ • (_ ◃ local_equivalence_inv_onecell_id y). Proof. refine (!(id2_left _) @ _ @ id2_right _). rewrite <- runitor_rinvunitor. rewrite !vassocr. apply maponpaths_2. rewrite local_equivalence_twocell_runitor. rewrite !vassocl. refine (!_). etrans. { do 2 apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite vcomp_rinv. rewrite lwhisker_id2. rewrite id2_left. apply idpath. } etrans. { apply maponpaths. rewrite !vassocr. rewrite vcomp_rinv. apply id2_left. } rewrite <- local_equivalence_inv_twocell_comp. rewrite rinvunitor_runitor. 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.