A note on a question of R. Pol concerning light maps

Let f :X→Y be an onto map between compact spaces such that all point-inverses of f are zero-dimensional. Let A be the set of all functions u :X→I=[0,1] such that u[f←(y)] is zero-dimensional for all y∈Y. Do almost all maps u :X→I, in the sense of Baire category, belong to A? Toruńczyk proved that the answer is yes if Y is countable-dimensional. We extend this result to the case when Y has property C.