$\mathbf{Rng}$ is not a subcategory of $\mathbf{Ab}$? I read in the book Categories and functors (Pareigis) that the category of rings Rng is not a subcategory of the category of abelian groups Ab.
I didn't understand the argument the author used: although each ring is an abelian group, the underlying structures of abelian group may coincide hence $\textrm{Obj}(\mathbf{Rng})\subseteq \textrm{Obj}(\mathbf{Ab})$ does not hold.
Of course the underlying structure of abelian groups may coincide even if the rings are different but I didn't understand why this imply rings are not a subcategory of abelian groups.
 A: What is meant in the book is that the forgetful functor $U : \mathsf{Rng} \to \mathsf{Ab}$ does not make $\mathsf{Rng}$ into a subcategory of $\mathsf{Ab}$, because two different rngs may have the same underlying abelian group. For example, consider $\mathbb{Z}$ with the usual operations, and the same $\mathbb{Z}$ but where $x \cdot y = 0$ for all $x$ and $y$. So $U$ is not an injection of objects, let alone an embedding of categories.
Now it is true that, even then, there could possibly be another functor $F : \mathsf{Rng} \to \mathsf{Ab}$ that makes $\mathsf{Rng}$ into a subcategory of $\mathsf{Ab}$, but this isn't what the author meant (in all likelihood). Here is what he wrote:

The corresponding abelian groups of two rings may coincide even if the
  rings do not coincide. The multiplication may be defined differently.

Which matches the explanation in the first paragraph. Finding a functor $F : \mathsf{Rng} \to \mathsf{Ab}$ that induces an injection on objects and hom-sets would be a harder question.
