Lemma yoneda_unit_component_is_modification (η : pstrans (representable B_is_univalent_2_1 X) F) : is_modification (yoneda_unit_component_mod_component_nat η). Proof. intros Z₁ Z₂ f. apply nat_trans_eq. { apply homset_property. } intro g ; cbn in g. cbn. unfold yoneda_unit_component_mod_component_nat_component, yoneda_to_presheaf_data_ob, presheaf_to_yoneda_ob_pstrans_nat_trans_data. cbn in f. etrans. { do 2 apply maponpaths. exact (nat_trans_eq_pointwise (pstrans_inv_comp_alt η g f) (id₁ X)). } cbn. rewrite !id_right. rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { apply functor_comp. } apply maponpaths_2. refine (!_). etrans. { apply maponpaths_2. refine (!(assoc _ _ _) @ _). apply maponpaths. etrans. { refine (!_). apply functor_comp. } apply maponpaths. etrans. { apply maponpaths_2. refine (!_). apply rinvunitor_triangle. } cbn. refine (vassocl _ _ _ @ _). etrans. { apply maponpaths. apply lassociator_rassociator. } apply id2_right. } etrans. { apply maponpaths_2. refine (!_). exact (pr21 (psnaturality_of η f) g (g · id₁ X) (rinvunitor g)). } cbn. etrans. { refine (!(assoc _ _ _) @ _). apply maponpaths. exact (nat_trans_eq_pointwise (vcomp_rinv (psnaturality_of η f)) (g · id₁ X)). } apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.Properties. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.PseudoFunctors.Yoneda. Require Import UniMath.Bicategories.PseudoFunctors.Representable. Require Import UniMath.Bicategories.Core.Examples.Image. Require Import UniMath.Bicategories.PseudoFunctors.Examples.CorestrictImage.