Definition right_biadj_preserves_inserters (HB₁ : is_univalent_2_1 B₁) (HB₂ : is_univalent_2_1 B₂) : preserves_inserters R. Proof. intros y₁ y₂ f g i Hi. use universal_inserter_cone_has_ump. intro x. use nat_iso_adj_equivalence_of_cats. - refine (biadj_left_hom R x i ∙ inserter_cone_functor i (L x) ∙ _). use dialgebra_equivalence_of_cats_functor. + exact (biadj_right_hom R x y₁). + exact (biadj_right_hom R x y₂). + exact (inserter_commute_nat_z_iso f x). + exact (inserter_commute_nat_z_iso g x). - exact (right_biadj_preserves_inserters_nat_trans i x). - exact (right_biadj_preserves_inserters_is_nat_z_iso i x). - use comp_adj_equivalence_of_cats. + use comp_adj_equivalence_of_cats. * exact (biadj_hom_equiv R x i). * apply (make_is_universal_inserter_cone HB₂ _ Hi). + use dialgebra_adj_equivalence_of_cats. * exact (adj_equivalence_of_cats_inv (biadj_hom_equiv R x y₁)). * exact (adj_equivalence_of_cats_inv (biadj_hom_equiv R x y₂)). 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Categories.Dialgebras. 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.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.Properties. Require Import UniMath.Bicategories.Morphisms.Properties.ClosedUnderInvertibles. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction. Require Import UniMath.Bicategories.PseudoFunctors.Preservation.Preservation. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.Limits.Inserters.