Good point, Jesus! It's not totally clear what it means for two categories to be the same or different. And somehow equivalence seems like the wrong notion, since two categories can be equivalent with different numbers of objects and morphisms. Assuming isomorphism is meant, it might first be helpful to count the isomorphism classes of monoids with four or fewer elements, since every object's endomorphism monoid (the monoid of morphisms from it to itself—see Puzzle 98) must be one of these.