  A new approach is suggested to characterize algebraically automorphisms of
the category of free algebras of a given variety. It gives in many cases an
answer to the problem set by the first of authors, if automorphisms of such a
category are inner or not. This question is important for universal algebraic
geometry [arXiv:math. GM\slash 0210187],[arXiv:math. GM\slash 0210194]. Most of
results are actually valid for arbitrary categories with a represented
forgetful functor.
