sage: Hom(ZZ, ZZ).category() # inherits from CommAdditiveGroups() now
Join of Category of endsets and Category of commutative additive groups and Category of homsets of Dedekind domains and euclidean domains and metric spaces
sage: # this used to be "Category of homsets of sets"
sage: class X(Parent):
....:     def _repr_(self): return "XXXXXX"
....: Hom(X(category=Sets()), ZZ).category()
Join of Category of commutative additive groups and Category of homsets of sets

sage: for R in [Zmod(10), QQ, NumberField(x^3 + 5, "a"), Qp(3), GF(2), GF(3), GF((7, 3))]:
....:     params = [3, 5] if R.characteristic() not in [2, 3] else [1, 0, 1, 3, 5]
....:     TestSuite(EllipticCurve(R, params)).run(skip="_test_not_implemented_methods")

sage: J = HyperellipticCurve(polygen(QQ)^5 + 1).jacobian()
sage: J.categories() # jacobians are abelian varieties
[Category of abelian varieties over Spectrum of Rational Field,
 Category of schemes over Rational Field,
 Category of schemes,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]
sage: J.point_homset().categories()
[Join of Category of commutative additive groups and Category of homsets of schemes over Rational Field,
 Category of commutative additive groups, ...
 Category of homsets of schemes over Rational Field, ...
 Category of sets with partial maps,
 Category of objects]

sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
....: P2.<x, y> = ProjectiveSpace(QQ, 1)
....: J = Jacobian_generic(Curve(y^2 - (x^3 + x)))
....: J in AbelianVarieties(QQ) # works for Jacobian_generic
True
sage: J = Jacobian_generic(Curve(y^2 - x^3))
....: J in AbelianVarieties(QQ) # only if the curve is projective smooth
False

sage: AbelianVarieties(QQ) # changed base scheme to be a scheme (lol?)
Category of abelian varieties over Spectrum of Rational Field
sage: AbelianVarieties(QQ) is AbelianVarieties(Spec(QQ)) # input is normalised
True