Categories are wrong for Hom/cartesian products of vector spaces

[videlec]

sage: cartesian_product([GF(3)^2, GF(7)^3])
The cartesian product of (
  Vector space of dimension 2 over Finite Field of size 3,
  Vector space of dimension 3 over Finite Field of size 7)
sage: _.category()
Category of Cartesian products of vector spaces with basis over
 (finite fields and subquotients of monoids and quotients of semigroups)

and

sage: Hom(GF(3)^2, GF(7)^3)
Set of Morphisms (Linear Transformations) from
  Vector space of dimension 2 over Finite Field of size 3
to
  Vector space of dimension 3 over Finite Field of size 7
sage: _.category()
Category of homsets of modules with basis over
 (finite fields and subquotients of monoids and quotients of semigroups)

see also: #16940

CC: @nthiery @tscrim

Component: categories

Issue created by migration from https://trac.sagemath.org/ticket/18309

[mkoeppe]

comment:1

The first one is now

sage: _.category()                                                                                                                
Join of Category of Cartesian products of vector spaces with basis over (finite enumerated fields and subquotients of monoids and quotients of semigroups) and Category of finite additive groups and Category of Cartesian products of finite enumerated sets

[mkoeppe]

comment:2

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.