Definition Ktriple_to_Monad : disp_psfunctor kleisli_triple_disp_bicat (monad bicat_of_univ_cats) (id_psfunctor bicat_of_univ_cats). Proof. use make_disp_psfunctor. - apply disp_2cells_isaprop_monad. apply univalent_cat_is_univalent_2. - exact (disp_locally_groupoid_monad bicat_of_univ_cats univalent_cat_is_univalent_2). - exact @unit_mu_kleisli. - exact @unit_mu_kleisli_functor. - abstract (cbn; intros x y f g α kx ky kf kg e; refine ((_,, (tt,, tt)),, tt); use nat_trans_eq; try apply homset_property; cbn; intro a; apply pathsinv0; apply z_iso_inv_on_left; rewrite assoc'; rewrite <- e; rewrite assoc; rewrite z_iso_after_z_iso_inv; apply pathsinv0; apply id_left). - abstract (intros x kx; refine ((_,, (tt,, tt)),, tt); unfold alg_disp_cat_2cell; use nat_trans_eq; try apply homset_property; intro a; cbn; rewrite !id_left; rewrite (bind_bind kx); rewrite (unit_bind kx); apply pathsinv0; apply (bind_unit kx)). - abstract (simpl; intros x y z f g kx ky kz kf kg; refine ((_,, (tt,, tt)),, tt); use nat_trans_eq; try apply homset_property; intro a; cbn; rewrite !id_left; rewrite !id_right; rewrite assoc'; rewrite (bind_bind kz); rewrite (unit_bind kz); rewrite (bind_unit kz); rewrite id_right; apply idpath). 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.