Gelfand numbers and widths

In general, the Gelfand widths c@?"n(T) of a map T between Banach spaces X and Y are not equivalent to the Gelfand numbers c"n(T) of T. We show that c@?"n(T)=c"n(T)(n@?N) provided that X and Y are uniformly convex and uniformly smooth, and T has trivial kernel and dense range.