Proposition local_equivalence_twocell_rwhisker {x y z : B₁} {f₁ f₂ : F x --> F y} (τ : f₁ ==> f₂) (g : F y --> F z) : local_equivalence_inv_twocell (τ ▹ g) • local_equivalence_inv_onecell_comp f₂ g = local_equivalence_inv_onecell_comp f₁ g • (local_equivalence_inv_twocell τ ▹ _). Proof. cbn -[psfunctor_comp]. use local_equivalence_eq_two_cell. rewrite !psfunctor_vcomp. rewrite !psfunctor_local_equivalence_inv_twocell. rewrite !vassocl. apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. rewrite !vassocl. apply idpath. } etrans. { rewrite !vassocr. do 4 apply maponpaths_2. rewrite rwhisker_vcomp. rewrite <- local_equivalence_counit_natural_alt. rewrite <- rwhisker_vcomp. apply idpath. } rewrite !vassocl. apply maponpaths. etrans. { rewrite !vassocr. rewrite vcomp_whisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite <- psfunctor_rwhisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite <- local_equivalence_counit_natural_alt. rewrite !vassocl. apply idpath. } apply maponpaths. rewrite <- !psfunctor_vcomp. apply maponpaths. rewrite local_equivalence_unit_natural_alt. 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.