Definition Monad_biequiv_Ktriple_unit_inv : disp_pstrans (disp_pseudo_id (monad bicat_of_univ_cats)) (disp_pseudo_comp (id_psfunctor bicat_of_univ_cats) (id_psfunctor bicat_of_univ_cats) (monad bicat_of_univ_cats) kleisli_triple_disp_bicat (monad bicat_of_univ_cats) Monad_to_Ktriple Ktriple_to_Monad) (linvunitor_pstrans (id_psfunctor bicat_of_univ_cats)). Proof. use make_disp_pstrans. - exact (disp_2cells_isaprop_monad bicat_of_univ_cats univalent_cat_is_univalent_2). - exact (disp_locally_groupoid_monad bicat_of_univ_cats univalent_cat_is_univalent_2). - intros. use make_cat_monad_mor. + simpl. cbn. use make_nat_z_iso. * use make_nat_trans. ** intro z. apply identity. ** abstract (intros z t f ; cbn ; rewrite id_left, id_right ; apply cat_monad_map_as_bind). * intros z. apply identity_is_z_iso. + intros z. apply id_right. + abstract (simpl ; intros X ; rewrite !id_left, id_right ; rewrite bind_unit ; rewrite id_left ; cbn ; unfold monad_bind ; rewrite functor_id ; rewrite id_left ; apply idpath). - abstract (intros ; use make_cat_monad_cell ; simpl ; intro z ; rewrite !id_left ; rewrite !id_right ; rewrite (functor_id f) ; rewrite id_left ; cbn ; rewrite <- assoc ; apply maponpaths ; refine (!_) ; refine (maponpaths (λ q, _ · q) (cat_monad_unit_bind _) @ _) ; apply id_right). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Projection. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.AlgebraMap. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Algebras. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Add2Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Monads. Require Import UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple. Require Import UniMath.Bicategories.DisplayedBicats.DispBuilders. Require Import UniMath.Bicategories.DisplayedBicats.DispTransformation. Require Import UniMath.Bicategories.DisplayedBicats.DispModification. Require Import UniMath.Bicategories.DisplayedBicats.DispBiequivalence. Require Import UniMath.Bicategories.Transformations.Examples.Unitality. Require Import UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses.