Not sure honestly, I can think of alternatives but I don't know if they're better. "Isomorphism of sets" is a fancy way of saying "bijection", and "unique set up to isomorphism" is the same as "unique cardinal" but as you say it's better to use terms that everyone knows. It might just be that "unique" being right at the beginning makes it easy to forget. Decoding the description is a fun part of the problem!