Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. Is $T$ $\kappa$-categorical?

If $|\mathcal L’| = \kappa = \aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $\aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.