It is important to note that, contrary to intuition, concreteness is not a property which a category may or may not satisfy, but rather a structure with which a category may or may not be equipped.
Similarly, any set with 4 elements can be given two non-isomorphic group structures: one isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}; the other isomorphic to \mathbb{Z}/4\mathbb{Z}.