proposition: Lemma algebra_as_coequalizer_ump_eq_lem_1 : # M hX · pr1 g = μ M X · pr1 g.
proof: Proof. refine (_ @ maponpaths pr1 (!p) @ _). - refine (maponpaths (λ z, z · _) _). simpl. rewrite (functor_comp M). refine (_ @ assoc (#M hX) _ _). refine (!(id_right _) @ _). apply maponpaths. refine (!_). apply Monad_law2. - refine (maponpaths (λ z, z · _) _). refine (_ @ id_left _). refine (maponpaths (λ z, z · _) _). apply functor_id. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Categories.HSET.All UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits UniMath.CategoryTheory.Monads.Monads