proposition: Lemma yoneda_unit_is_nat_trans : is_nat_trans (functor_identity (hom_data (representable B_is_univalent_2_1 X) F)) (yoneda_to_presheaf ∙ presheaf_to_yoneda) yoneda_unit_component_mod.
proof: Proof. intros η₁ η₂ m. apply modification_eq. intros Z. apply nat_trans_eq. { apply homset_property. } intros g ; cbn in g. cbn. unfold yoneda_unit_component_mod_component_nat_component, yoneda_to_presheaf_data_ob, presheaf_to_yoneda_mor_modification_nat_trans_data, yoneda_to_presheaf_data_mor. refine (!_). etrans. { rewrite <- assoc. apply maponpaths. exact (!(nat_trans_eq_pointwise (mod_inv_naturality_of m X Z g) (id₁ X))). } simpl. rewrite !assoc. apply maponpaths_2. exact (pr2 ((m : modification _ _) Z) _ _ (rinvunitor g)). 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