TL;DR: compact-open topology for Homs of locales?

Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.

For two locales, $A$ and $B$, is there a nice way to make an internal (to $\mathcal{L}$ or to $\mathcal{L}$) Hom out of $Hom_{\mathcal{L}}(A, B)$? That is, an exponential object $B^A$ in $\mathcal{L}$ or at least $\mathcal{Loc}$. Will it work to take the compact-open topology on the mapping space $Hom_{\mathcal{Top}}(X, Y)$ for any pair of topological spaces such that the frame of opens on $X$ is $A$ and that of $Y$ is $B$?