$C_1 \subseteq C_2$ implies $C_1^A\subseteq C_2^A$?
I've given a caveat that one shouldn't make this implication blindly and it shall be justified.
I can think of examples such that $C_1^A \subsetneq C_2^A$ and examples such that $C_1^A = C_2^A$.
but could it be that $C_2^A \subset C_1^A$? I can't see how.
So what's the point of this caveat?