Formalize "naturality" of a functor? Wikipedia states:

There is a natural functor from Ring to the category of groups, $\mathrm{Grp}$, which sends each ring $R$ to its group of units $U(R)$ and each ring homomorphism to the restriction to $U(R)$.

I have never heard of a "natural functor" before and havent found about it online. Is this simply a natural transformation in the category of categories? (Where functors are morphisms)
If so, how do we formalize the notion that the abovementioned functor is natural?
 A: A "Natural" functor is typically a functor that arises from a natural mathematical situation,
ie. $U:\textbf{Ring} \rightarrow \textbf{Grp}$ described as earlier, $F:\textbf{Ring}\rightarrow \textbf{Ab}$ sends a ring to its underlying abelian group, and morphisms to abelian group homomorphisms.
Various constructions can be described in terms of functors, which was the motivation for Category Theory to begin with.
A: I'm writing this as an answer because it is too large to be a comment. In your specific example there might be various notions of "naturality" under consideration.
First of all, you might consider the category of group objects in the monoidal category $(Set,\times)$, which is simply the category of groups $Grp$. There's a "natural" functor from the category of $(Set,\times)$-group objects to $Set$ which simply forgets the group object structure. This notion of naturality works in many other examples, such as monoid objects in a monoidal category, Lie algebra objects in a symmetric monoidal category, etc. Repeating this process in $Grp$ we obtain the category of $(Grp,\times)$-group objects, which is simply $Ab$, the category of abelian groups. Again there is a "natural" (in the sense mentioned before) functor $Ab \to Grp$. That yields us a "natural" composition $Ab \to Set$. Read more about group objects here.
Now, we can consider the categories of monoid objects in the monoidal categories $(Ab,\otimes)$ and $(Set,\times)$. That gives us respectively the categories of rings and ordinary monoids. The functor $Ab \to Set$ mentioned previously induces "naturally" a functor $Ring \to Mon$. Read more about that here.
Finally, there's the functor $Mon \to Grp$ which sends each monoid to its group of "units" and each monoid homomorphism to its restriction, but I'm unable to find any references on this kind of construction.
