Strongly alg-universal semigroup varieties

Abstract. If $\Bbb V$ is a semigroup variety such that every monoid is isomorphic to the endomorphism monoid of a semigroup from $\Bbb V$, then any full category $\Bbb A$ of algebras has a full embedding into $\Bbb V$ carried by an endofunctor of the category $\Bbb S\Bbb E\Bbb T$ of all sets and mappings.