proposition: Lemma algebra_as_coequalizer_ump_eq : hX · (η M X · pr1 g) = # M (η M X · pr1 g) · pr12 Y.
proof: Proof. rewrite functor_comp. rewrite !assoc'. refine (!_). etrans. { apply maponpaths. exact (!(pr2 g)). } rewrite !assoc. rewrite (@Monad_law2 _ M). rewrite id_left. refine (!_). apply algebra_as_coequalizer_ump_eq_lem_2. 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