  Following ideas of Lawvere and Linton we prove that classical varieties are
precisely the exact categories with a varietal generator. This means a strong
generator which is abstractly finite and regularly projective. An analogous
characterization of varieties of ordered algebras is also presented. We work
with order-enriched categories, and introduce the concept of subexact category
and subregular projective (corresponding naturally to the ordinary case).
Varieties of ordered algebras are precisely the subexact categories with a
subvarietal generator. This means a strong generator which is abstractly finite
and subregularly projective.
