Let $X_{\mu}$, $\mu \in \Lambda$ are metric spaces i.e. objects of $MET$ which contain more than one element. The problem is to prove that their (categorical) product exists if and only if $\Lambda$ is at most countable.
(I didn't do anything myself yet, I just don't know where to begin)
Hints:
$\Leftarrow$: If $\Lambda$ is countable, you can explicitly write down a metric for the product of the $X_\mu$ in the category of all topological spaces. It is therefore also a product in the category of metric spaces.
$\Rightarrow$: Compare the hypothetical product in the category of metric spaces to the one in the category of all topological spaces. Show that they share the same elements and that one is finer than the other. Finally, consider the fact that in any metrizable topological space every point has a countable neighbourhood base.