On cardinality questions concerning automaton mappings

Let F+(X) be the set of words of positive length over a finite set X. By an automaton mapping (over (X,Y)) we understand a mapping of F+(X) into a finite set Y where |Y|?1). The family of all mappings over (X,Y) may be considered as an infinite automaton U having 2? states. U has at most 2^{2?} subautomata and at most 2? countable subautomata. We show that these bounds are actually attained.