Let me elaborate on unknown (google)'s comment. If $\text{Rng}$ denotes the category of unital rings and $\text{Grp}$ denotes the category of groups, then it turns out that "group of units" is a functor $\text{Rng} \to \text{Grp}$. In fact, it's representable; the group of units of a ring $R$ can be identified with $\text{Hom}(\mathbb{Z}[x, x^{-1}], R)$ (the set of all ring homomorphisms from $\mathbb{Z}[x, x^{-1}]$ to $R$), where the group structure on the Hom-set comes from a Hopf algebra structure on $\mathbb{Z}[x, x^{-1}]$. And covariant representable functors preserve limits, including binary products.
Closely related to representability (although I've never been clear on the precise relationship) is the fact that "group of units" has a left adjoint $\text{Grp} \to \text{Rng}$ which constructs the group ring of a group. And functors which have left adjoints preserve limits.
Presumably many other examples can be handled in a similar vein, although I can't think of any right now, so you'll have to give me more examples.