proposition: Lemma yoneda_unit_is_inverses (g : pstrans (representable B_is_univalent_2_1 X) F) (Z : B) (Y : Z --> X) : is_inverse_in_precat (# (g Z : _ ⟶ _) (rinvunitor Y) · pr1 ((psnaturality_of g Y) ^-1) (id₁ X)) ((pr11 (psnaturality_of g Y)) (id₁ X) · # (g Z : _ ⟶ _) (runitor Y)).
proof: Proof. split. - rewrite <- !assoc. etrans. { apply maponpaths. etrans. { rewrite assoc. apply maponpaths_2. exact (nat_trans_eq_pointwise (vcomp_linv (psnaturality_of g Y)) (id₁ X)). } apply id_left. } refine (!(functor_comp _ _ _) @ _ @ functor_id (g Z) _). apply maponpaths. exact (rinvunitor_runitor Y). - rewrite <- !assoc. etrans. { apply maponpaths. etrans. { rewrite assoc. apply maponpaths_2. refine (!(functor_comp (g Z) _ _) @ _ @ functor_id (g Z) _). apply maponpaths. apply runitor_rinvunitor. } apply id_left. } exact (nat_trans_eq_pointwise (vcomp_rinv (psnaturality_of g Y)) (id₁ X)). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Examples.OpMorBicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Morphisms.Properties UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.UnivalenceOp UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Properties UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.PseudoFunctors.Yoneda UniMath.Bicategories.PseudoFunctors.Representable UniMath.Bicategories.Core.Examples.Image UniMath.Bicategories.PseudoFunctors.Examples.CorestrictImage