= Solution
A <first-order theory>[theory] is $\kappa$-categorical if it has a <first-order model>[model] of <cardinality> $\kappa$ and any two <first-order model>[models] of that <cardinality> are isomorphic. The existence clause prevents vacuous categoricity at cardinalities where there is no <first-order model>[model].
To show that <infinite-cardinal categoricity forces completeness>, suppose $T$ were incomplete. Then some sentence $\sigma$ has both $T+\sigma$ and $T+\neg\sigma$ satisfiable, by completeness of <first-order logic>. Their <first-order model>[models] must be infinite because $T$ has no finite <first-order model>[models]. For an infinite $\kappa\ge|L|$, the upward and downward <Löwenheim-Skolem theorems> produce a <first-order model>[model] of each extension of <cardinality> exactly $\kappa$: pass to an elementary submodel of size at most $\max(|L|,\aleph_0)$ and then to the desired size, or reduce a larger <first-order model>[model] to that size. Categoricity makes these two <first-order model>[models] isomorphic, but one satisfies $\sigma$ and the other its <negation>. <Contradiction>. Therefore \b[$T$ is complete]. If language size is defined without automatically including $\aleph_0$, the infinite-cardinal qualification is explicit; the assumed existence of a $\kappa$-sized <first-order model>[model] and absence of finite <first-order model>[models] already force it here.
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