It was very often mentioned that the notion of equivalence of two categories was more
preferable than the notion of
the automorphism for category theory itself and for most of its applications.
But it is a very natural question in which cases there are no equivalences very
different from automorphisms, i.e. they are not isomorphic to any automorphism.
We call such autoequivalences proper autoequivalences. This question
was set for the first time by B.I. Plotkin in connection with his studying the universal
algebraical geometry. Plotkin's question concerned mainly some special categories namely
the categories
of free universal algebras of a variety over
finite subsets of a fixed infinite set . But the answer
led out this special case.

**Theorem 1.** *Let be an arbitrary small (it is enough for our aim) category.
For every object of , let denote the set of all -objects,
which are isomorphic to . An autoequivalence is isomorphic to an automorphism
if and only if for every two objects and such that
the sets
and are of equal cardinality.*

A small category has no proper autoequivalences if and only if
for every automorphism of a skeleton
of this category and for every two objects and of such that ,
the sets and are of equal cardinality. This gives the following answer to Plotkin's
question: in the categories of the kind
there are no proper autoequivalences.
The proof of Theorem 1 uses the Axiom of Choice but in some simple cases it is possible to
remove references to this axiom, for example for the category
if
the set is denumerable.

Now let be an arbitrary category supplied with a faithful functor
with the following condition:

(*) Let and be two objects of this category such that and have the same
cardinality. Then for every bijection
there exists an unique object
of and an isomorphism
such
that and
.
For such categories a sufficient condition is
given for an autoequivalence to be isomorphic to an automorphism. In particular, this condition
is satisfied if the functor is represented by an object such
that is isomorphic to . Thus every full subcategory of containing singleton
does not contain any proper autoequivalences.

Let be a variety of universal algebras and be the corresponding category.
Let
be the forgetting functor. It is clear that condition (*) is satisfied.
Moreover, the functor is represented by the free algebra in over an
one-element set. Thus if is isomorphic to for an autoequivalence ,
then is isomorphic to an automorphism. Indeed, the same fact is valid for
every full subcategory of if it satisfies the condition
(*) and contains a free algebra over a singleton. Hence we have

**Theorem **
*Let be the category of all free semigroups, or of all free inverse semigroups
or of all free groups. Then there are no proper autoequivalences in .*

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*.