= Solution
A model of cardinality $\kappa$ is determined up to isomorphism by the unordered pair of cardinalities of its two equivalence classes. At cardinality $\aleph_0$, both infinite classes must be countable, so there is one isomorphism type. Thus $T_1$ is <categorical theory> in $\aleph_0$.
For every uncountable $\kappa$, a model with class sizes $(\aleph_0,\kappa)$ is not isomorphic to one with sizes $(\kappa,\kappa)$. Hence $T_1$ is not $\kappa$-categorical for any uncountable $\kappa$; it has no finite models.
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