Let $X$ be a compact Hausdorff space. Then if $f: A \to B$ is a map between discrete spaces, the induced map $f^\ast: X^B \to X^A$ is proper.
Question: Are there other classes of map $f: A \to B$ such that $f^\ast: X^B \to X^A$ is proper for every compact Hausdorff space $X$?
Notes:
Part of what I find puzzling about the state of affairs is that when $A$ and $B$ are discrete, we don't need to assume that $f$ itself is proper in order to conclude that $f^\ast$ is proper.
Of course, in order for the question to make sense, we should probably assume that $A$ and $B$ are exponentiable (locally compact Hausdorff is fine). For all I know, though, it may be the case that there are spaces $A$ which are not exponentiable in general, but for which $X^A$ exists for every compact Hausdorff space; the question would make sense for $A,B$ of this form as well.
I'm not certain that restricting $X$ to be compact Hausdorff is the most interesting thing to do, so I'd be interested in this question for other $X$'s as well.
