  In the first part of our paper (Sections 1, 2 and 3) we reprove results of B.
Plotkin, G. Zhitomirski. On automorphisms of categories of free algebras of
some varieties, Journal of Algebra, 306:2, (2006), 344 -- 367 for the case of
many-sorted algebras. In the second part of our paper (Section 4) we apply the
results of the first part to the universal algebraic geometry of many-sorted
algebras and refine and reprove results of B. Plotkin, Algebras with the same
(algebraic) geometry, Proceedings of the International Conference on
Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of
P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, 242
(2003), 127 -- 207 and A. Tsurkov, Automorphic equivalence of algebras,
International Journal of Algebra and Computation. 17:5/6, (2007), 1263 -- 1271
for these algebras. In the third part of this paper (Section 5) we consider
some varieties of many-sorted algebras. We prove that automorphic equivalence
coincide with geometric equivalence in the variety of the all actions of
semigroups over sets and in the variety of the all automatons. We also consider
the variety of the all representations of groups and the all representations of
Lie algebras. For both these varieties we give an examples of the
representations which are automorphically equivalent but not geometrically
equivalent.
