A question on characterization of concrete categories

Question

All objects of a category are subsets of a set. Is the category (isomorphic to) a concrete category?

Answer 1

If the objects are all members of $\mathcal P(X)$ for some $X$ and additionally the hom-sets are sets, then the category is small, and every small category is concretizable. (The forgetful functor can be taken to map each object to the set of morphisms ending at that object).

Conversely, if some $\mathrm{Hom}(A,B)$ is a proper class, then the category clearly cannot be concretizable.