proposition: Lemma inv_monad_mor_natural_pointwise {C₁ C₂ : univalent_category} {F : C₁ ⟶ C₂} {M₁ : monad bicat_of_univ_cats C₁} {M₂ : monad bicat_of_univ_cats C₂} (FF : M₁ -->[F] M₂) (X : C₁) : inv_from_z_iso (monad_mor_natural_pointwise FF X) = CompositesAndInverses.nat_z_iso_to_pointwise_z_iso (monad_mor_nat_z_iso FF) X.
proof: Proof. refine (!_). apply inv_z_iso_unique'. unfold precomp_with. cbn. apply (nat_trans_eq_pointwise (vcomp_linv (monad_mor_natural FF)) X). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.DisplayedCats.Core UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispAdjunctions UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.PseudoFunctors.Examples.Identity UniMath.Bicategories.PseudoFunctors.Examples.Composition UniMath.Bicategories.PseudoFunctors.Examples.Projection UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Transformations.Examples.AlgebraMap UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.PseudoFunctors.Biequivalence UniMath.Bicategories.Core.Unitors UniMath.Bicategories.DisplayedBicats.DispPseudofunctor UniMath.Bicategories.DisplayedBicats.Examples.Algebras UniMath.Bicategories.DisplayedBicats.Examples.Add2Cell UniMath.Bicategories.DisplayedBicats.Examples.Prod UniMath.Bicategories.DisplayedBicats.Examples.FullSub UniMath.Bicategories.DisplayedBicats.Examples.Monads UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple UniMath.Bicategories.DisplayedBicats.DispBuilders UniMath.Bicategories.DisplayedBicats.DispTransformation UniMath.Bicategories.DisplayedBicats.DispModification UniMath.Bicategories.DisplayedBicats.DispBiequivalence UniMath.Bicategories.Transformations.Examples.Unitality UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple UniMath.CategoryTheory.Equivalences.CompositesAndInverses