Definition bicat_of_univ_cats_has_kleisli_ump_1 : has_kleisli_ump_1 m bicat_of_univ_cats_has_kleisli_cocone. Proof. intro q. pose (F := mor_of_kleisli_cocone _ q). pose (γ := cell_of_kleisli_cocone _ q). pose (p₁ := nat_trans_eq_pointwise (kleisli_cocone_unit _ q)). assert (p₂ : ∏ (x : C), pr1 γ (M x) · pr1 γ x = # (pr1 F) (μ M x) · pr1 γ x). { intro x. pose (p₂ := nat_trans_eq_pointwise (kleisli_cocone_mult _ q) x). cbn in p₂. rewrite !id_left in p₂. exact p₂. } use make_kleisli_cocone_mor. - exact (functor_from_univalent_kleisli_cat M F γ p₁ p₂). - exact (functor_from_univalent_kleisli_cat_nat_trans M F γ p₁ p₂). - abstract (use nat_trans_eq ; [ apply homset_property | ] ; intro x ; refine (_ @ maponpaths (λ z, z · _) (!(id_left _))) ; exact (functor_from_univalent_kleisli_cat_eq M F γ p₁ p₂ x)). - use is_nat_z_iso_to_is_invertible_2cell. exact (functor_from_univalent_kleisli_cat_nat_trans_is_z_iso M F γ p₁ p₂). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Categories.EilenbergMoore. Require Import UniMath.CategoryTheory.Categories.KleisliCategory. Require Import UniMath.CategoryTheory.CategorySum. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Monads.KleisliCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Reindexing. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Subcategory.Core. Require Import UniMath.CategoryTheory.Subcategory.Full. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.Examples.MonadsLax. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.Examples.MorphismsInBicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Colimits.Coproducts. Require Import UniMath.Bicategories.Colimits.Extensive. Require Import UniMath.Bicategories.Colimits.KleisliObjects. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.PullbackFunctions. Require Import UniMath.Bicategories.Limits.CommaObjects. Require Import UniMath.Bicategories.Limits.Examples.BicatOfUnivCatsLimits. Require Import UniMath.Bicategories.Monads.Examples.MonadsInBicatOfUnivCats. Require Import UniMath.Bicategories.Monads.Examples.MonadsInOp1Bicat.