Confusion of additive and multiplicative notations. 
If gcd$(m,n)=1$, then $\phi(mn)=\phi(m)\phi(n)$.

My text book write that $f:\Bbb Z/_{(mn)}\Bbb Z\to\Bbb  Z/_m\Bbb Z \times \Bbb Z/_n\Bbb Z$ by $f([a]_{mn})=([a]_m,[a]_n)$ is an isomorphism.
So far, I think he discusses the additive group.
But when he claims that $f(U(_{mn}\Bbb Z))=U(_m\Bbb Z)\times U(_n\Bbb Z)$, he using the multiplication on  $\Bbb Z/_{(mn)}\Bbb Z$ and I'm confused that whether changing operations is alright or not. If he want to use the multiplication on $\Bbb Z/_{(mn)}\Bbb Z$, why don't he just write that $f:(\Bbb Z/_{(mn)}\Bbb Z)^{\times}\to(\Bbb  Z/_m\Bbb Z)^\times \times (\Bbb Z/_n\Bbb Z)^\times$ although  $(\Bbb Z/_n\Bbb Z)^\times$ and $(\Bbb  Z/_m\Bbb Z)^\times$ may not be group.
Hope someone can help me to understand the notation here.

For a ring $R$, the group of units is denoted either by $U(R)$ or by $R^{\times}$. In your example the ring is $R=\Bbb Z/m\Bbb Z$ fo some integer $m$. So $(\Bbb Z/m\Bbb Z)^{\times}$ is certainly a group.
