Definition make_cat_monad (C : univalent_category) (M : C ⟶ C) (η : functor_identity C ⟹ M) (μ : M ∙ M ⟹ M) (lid : ∏ (X : C), #M (η X) · μ X = id₁ (M X)) (rid : ∏ (X : C), η (M X) · μ X = id₁ (M X)) (massoc : ∏ (X : C), μ (M X) · μ X = #M (μ X) · μ X) : monad bicat_of_univ_cats C. Proof. use make_monad. - exact M. - exact η. - exact μ. - abstract (use nat_trans_eq; try apply homset_property; intros X ; cbn; apply lid). - abstract (use nat_trans_eq; try apply homset_property; intros X ; cbn; apply rid). - abstract (use nat_trans_eq; try apply homset_property; intros X ; cbn; rewrite id_left; apply massoc). 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.Univalence. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. 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.Examples.Projection. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.AlgebraMap. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. 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.Sigma. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses.