Proposition local_equivalence_twocell_lunitor {x y : B₁} (f : F x --> F y) : lunitor (local_equivalence_inv_onecell f) = ((local_equivalence_inv_onecell_id x)^-1 ▹ _) • (local_equivalence_inv_onecell_comp _ _)^-1 • local_equivalence_inv_twocell (lunitor f). Proof. cbn -[psfunctor_id psfunctor_comp]. use local_equivalence_eq_two_cell. rewrite <- rwhisker_vcomp. rewrite !psfunctor_vcomp. rewrite !psfunctor_local_equivalence_inv_twocell. rewrite !vassocl. refine (!_). etrans. { do 7 apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. } rewrite psfunctor_F_lunitor. rewrite <- vcomp_lunitor. rewrite !vassocr. apply maponpaths_2. rewrite !vassocl. use vcomp_move_L_pM. { is_iso. } cbn -[psfunctor_id psfunctor_comp]. etrans. { rewrite !vassocr. rewrite psfunctor_rwhisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. rewrite psfunctor_rwhisker. rewrite !vassocl. apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite local_equivalence_triangle1. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite vcomp_rinv. rewrite id2_left. apply idpath. } etrans. { do 2 apply maponpaths. rewrite <- vcomp_whisker. rewrite !vassocl. rewrite lwhisker_vcomp. rewrite vcomp_rinv. rewrite lwhisker_id2. apply id2_right. } rewrite !rwhisker_vcomp. apply maponpaths. rewrite local_equivalence_counit_natural. rewrite !vassocr. rewrite local_equivalence_triangle1. apply id2_left. 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.