proposition: Proposition algebra_as_coequalizer_ump_unique : isaprop (∑ φ, algebra_as_coequalizer_arr · φ = g).
proof: Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. apply homset_property. } use subtypePath. { intro. apply homset_property. } refine (!(id_left _) @ _ @ id_left _). etrans. { apply maponpaths_2. exact (!(monad_algebra_unit hX)). } refine (!_). etrans. { apply maponpaths_2. exact (!(monad_algebra_unit hX)). } rewrite !assoc'. apply maponpaths. refine (!_). exact (maponpaths pr1 (pr2 φ₁ @ !(pr2 φ₂))). 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