Summary: We consider varieties of recognizable subsets of many-sorted finitely generated free algebras over a given variety, varieties of congruences of such algebras, and varieties of finite many-sorted algebras. A variety theorem that establishes bijections between the classes of these three types of varieties is proved. For this, appropriate notions of many-sorted syntactic congruences and algebras are needed. Also an alternative type of varieties is considered where each subset consists of elements of just one sort.