proposition: Lemma monoid_equiv_is_z_iso (X Y : ob monoid_category) (f : monoidiso X Y) : @is_z_isomorphism monoid_precategory X Y (monoidfunconstr (pr2 f)).
proof: Proof. exists (monoidfunconstr (pr2 (invmonoidiso f))). split. - use monoidfun_paths. use funextfun. intros x. use homotinvweqweq. - use monoidfun_paths. use funextfun. intros y. use homotweqinvweq. Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.UnivalenceAxiom UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Free_Monoids_and_Groups UniMath.Combinatorics.Lists UniMath.Algebra.IteratedBinaryOperations UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Adjunctions.Core