  Let V be a variety of algebras of some type. An interest to describing
automorphisms of the category C of finitely generated free V-algebras was
inspired in connection with development of universal algebraic geometry founded
by B. Plotkin. There are a lot of results on this subject. A common method of
getting such results was suggested and applied by B. Plotkin and the author (B.
Plotkin , G. Zhitomirski, On automorphisms of categories of free algebras of
some varieties, J. Algebra 306(2) (2006) 344-367). The method is to find all
terms in the language of a given variety which determine such V-algebras that
are isomorphic to a given C-algebra and have the same underlying set with it.
But this method can be applied only to automorphisms which take all objects to
isomorphic ones. The aim of the present paper is to suggest another method that
is free from the mentioned restriction. This method is based on two main
theorems. Let C be a category supplied with a faithful representative functor
to the category of sets. Theorem 1 gives a general description of automorphisms
of C, using a new notion of a quasi-inner automorphism. Theorem 5 shows how to
obtain the full description of automorphisms of the category of finitely
generated free V-algebras. The investigation ends with two examples. The first
of them shows the preference of our method in a known situation (the variety of
all semigroups) and the second one demonstrates obtaining new results (the
variety of all modules over arbitrary ring with unit and the consequence for
the rings without zero divisors).
