Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 144 / 3 / c / Solution

No. Let $L$ have unary predicates $P,Q$ and a unary function $f$. Let $T$ say that $P$ and $Q$ partition the universe into two infinite sets and that $f$ is an involution mapping $P$ bijectively onto $Q$. This is a complete theory. In every infinite cardinal $\kappa$, a model of total size $\kappa$ has $|P|=|Q|=\kappa$, and any two such bijections are isomorphic. Hence $T$ is $\kappa$-categorical for every infinite $\kappa$.

Take $L_0=\{P,Q\}$. The reduct theory merely says that $P$ and $Q$ are two infinite parts. At any uncountable $\kappa$, it has one model with $|P|={\aleph }_0$ and $|Q|=\kappa$, and another with $|P|=|Q|=\kappa$. They are not isomorphic. Therefore $T_0$ is not $\kappa$-categorical, giving the reduct need not preserve uncountable categoricity counterexample.