  With any even Hecke symmetry R (that is a Hecke type solution of the
Yang-Baxter equation) we associate a quasitensor category. We formulate a
condition on R implying that the constructed category is rigid and its
commutativity isomorphisms R_{U,V} are natural. We show that this condition
leads to rescaling of the initial Hecke symmetry. We suggest a new way of
introducing traces as properly normalized categorical morphisms End(V) --> K
and deduce the corresponding normalization from categorical dimensions.
