I asked this question (and have received an answer) at MathOverflow.
Now a little more difficult question:
Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is defined by the formula: $(f\times^{C} g) a = g\circ a \circ f^{-1}$ (for every binary relation $a$ on $\mho$).
Suppose $f_0$, $f_1$, $g_0$, and $g_!$ are non-empty. Knowing $f_0\times^{C} g_0\subseteq f_1\times^{C} g_1$, can we infer $f_0\subseteq f_1$ and $g_0\subseteq g_1$?
I am especially interested in short elegant proofs, because I am working on generalizing this.