Exponential of monos over terminal object
Solution 1:
Yes, $V$ may be arbitrary. So long as $U$ is subterminal $U^V$ will also be subterminal. In other words, you have shown that the class of subterminal objects is an exponential ideal.
Yes, $V$ may be arbitrary. So long as $U$ is subterminal $U^V$ will also be subterminal. In other words, you have shown that the class of subterminal objects is an exponential ideal.