Define a class as small if there is no injection from $V$ into the class. Suppose then that $a$ is small and let $F: a \rightarrow b$ be a surjection. Is $b$ small?
If we assume Choice, the answer is yes, since, if there is a surjection from $a$ to $b$, then there is an injection from $b$ to $a$, and then, if $b$ is not small, $a$ is not either. But what if we don't assume Choice (and, moreover, do not assume Replacement)? Can we still prove that $b$ is small?
(Incidentally, I'm assuming we are working within $\mathrm{Z}$, i.e. Zermelo's set theory, which is basically $\mathrm{ZFC}$ minus Replacement and Choice. The class terms are all eliminable as in Levy's Basic Set Theory.)