Definition coevaluation_pstrans_data: pstrans_data (id_psfunctor CAT) (comp_psfunctor R L). Proof. use make_pstrans_data. + intro A. apply coevaluation_functor. + intros A A' F. apply nat_z_iso_to_invertible_2cell. use make_nat_z_iso. * use make_nat_trans. -- intro a. cbn in a. cbn. ++ use make_nat_trans. ** intro b. apply identity. ** intros b b' f. cbn. apply pathsdirprod. --- do 2 rewrite id_right. apply functor_id. --- rewrite id_left. apply id_right. -- intros a a' g. cbn. apply (nat_trans_eq (homset_property (productwithfixedelement _ _))). cbn. intro b. apply pathsdirprod. ++ rewrite id_left. apply id_right. ++ apply idpath. * intro a. apply nat_trafo_z_iso_if_pointwise_z_iso. intro b. cbn. cbn in F, a. set (aux := identity(C:=pr1(productwithfixedelement _ _)) (make_dirprod (F a) b)). change (is_z_isomorphism aux). apply identity_is_z_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.UnitorsAndAssociatorsForEndofunctors. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.Modifications.Modification. Import Bicat.Notations.