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. 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.
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.