In the wikipedia article on Model Theory in the section on Categoricity it is said that:
As observed in the section on first-order logic, first-order theories cannot be categorical, i.e. they cannot describe a unique model up to isomorphism, unless that model is finite.
And in the linked article on first order logic they refer to the Löwenheim-Skolem-Theorem, which states that every first order sentence that has an infinite model, already has a countable model. But what if the only model is already countable, then Löwenheim-Skolem does not give anything new, so why then could this sentence not be categorical?