The second depends on the context feature only but not on f .
if we fix the context feature C to select the function f , using Corr(P (Y = 1|X), f (P (C = 1|X))) 2 (correlation with the gold standard P (Y = 1|X)) and Corr(P (Y = 1|P (C = 1|X)), f (P (C = 1|X))) 2 (correlation with the "silver" standard P (Y = 1|P (C = 1|X))) produce the same result.