Is multiplication modulo $10$ isomorphic to addition modulo $4$?
$U(10) = \{1,3,7,9\}$, the identity is $1$, it is a cyclic group of order $4$, with generator $3$.
$\Bbb Z_4 = \{0,1,2,3\}$, the identity is $0$, it is a cyclic group of order $4$
\begin{gather*} 1 \mapsto 0 \\ 3 \mapsto 1 \\ 9 \mapsto 2 \\ 7 \mapsto 3 \end{gather*} (Is this map sufficient to show multiplication mod $10$ is isomorphic to addition mod $4$?
As IBWiglin says in the comments, you need to check that this map $f$ is a group homomorphism. That means for every $a,b \in U(10)$ you need to check $f(ab) = f(a) + f(b)$. You can easily check that it is true here.
More generally if $a,b$ are coprime numbers, then $U(ab) \cong U(a) \times U(b)$ and $U(p)$ is isomorphic to $\mathbb{Z}_{p-1}$ for every prime number $p$. So in your case: $$\begin{align*} U(10) & \cong U(2) \times U(5) \\ & \cong \mathbb{Z}_1 \times \mathbb{Z}_4 \\ & \cong \mathbb{Z}_4 \end{align*}$$