1Note that CAT cannot be an object of CAT, which would lead to Russel’s paradox. A “smallness” condition is needed to avoid this. One usually requires that the objects of CAT are categories that have sets of objects, while the objects of CAT itself form a proper class. In type theory, this would correspond to saying that the objects of CAT live in the first universe, while CAT itself lives in the second.
For the purpose of this exercise, you can safely ignore this issue.