Definition right_biadj_preserves_bifinal : preserves_bifinal R. Proof. intros y Hy. use is_bifinal_repr_to_is_bifinal. intro x. use nat_iso_adj_equivalence_of_cats. - exact (biadj_left_hom R x y ∙ functor_to_unit _). - use make_nat_trans. + exact (λ _, idpath _). + abstract (intros f g α ; apply isapropunit). - intros f. use tpair; cbn. + apply idpath. + abstract (split ; apply idpath). - use comp_adj_equivalence_of_cats. + exact (biadj_hom_equiv R x y). + exact (is_bifinal_to_is_bifinal_repr Hy (L x)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction. Require Import UniMath.Bicategories.PseudoFunctors.Preservation.Preservation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Limits.Final.