= Solution
A <complete theory> is $\kappa$-categorical when it has a model with underlying <cardinality> $\kappa$ and any two of its models of that <cardinality> are <isomorphic structures>. Thus categoricity means uniqueness up to isomorphism at the specified <cardinal>, not a unique underlying set or a unique isomorphism.
For the infinite <cardinals> relevant here, the existence clause is automatic from the stated infinite model and the <Löwenheim-Skolem theorem>. One may equivalently require exactly one isomorphism class of models of size $\kappa$.
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