proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Properties