Definition make_cat_monad_mor {C D : univalent_category} {mx : monad bicat_of_univ_cats C} {my : monad bicat_of_univ_cats D} {F : C ⟶ D} (mf_nat : nat_z_iso (monad_endo mx ∙ F) (F ∙ monad_endo my)) (mfη : ∏ (X : C), # F (pr1 (monad_unit mx) X) · mf_nat X = pr1 (monad_unit my) (F X)) (mfμ : ∏ (X : C), # F (pr1 (monad_mu mx) X) · mf_nat X = mf_nat (pr1 (monad_endo mx) X) · # (pr1 (monad_endo my)) (mf_nat X) · pr1 (monad_mu my) (F X)) : mx -->[F] my. Proof. use make_monad_mor. - apply nat_z_iso_to_invertible_2cell. exact mf_nat. - abstract (use nat_trans_eq; try apply homset_property; intros X ; cbn; do 2 rewrite id_left; apply mfη). - abstract (use nat_trans_eq; try apply homset_property; intros X ; cbn; rewrite id_left, !id_right; apply mfμ). 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.