modular arithmetic - Proving isomorphism on additive group $(Bbb{Z}_4,+)$ and multiplicative group $(Bbb{Z}_5^*, times)$

The question I'm having problems with involves proving the above groups are isometric. Therefore, I have to prove they are bijective (1-1, and onto) and homomorphic. I have done the group operations table for each and come up with:

Set $A = (\mathbb{Z}_4,+) = {0,1,2,3}$

Set $B = (\Bbb{Z}_5^*, \times) = {1,2,3,4}$

So $B$, can be mapped from $A$ with the function: $\alpha(x)=x+1$

We can see no element of $B$ is the image of more than one element in $A$, therefore we have proven a 1-1 correspondence, and onto.

Now I have to prove they are isomorphic by exhibiting a 1-1 corresponds α between their elements such that:

$a+b \equiv c\ (\text{mod}\ 4)$ if and only if $\alpha(a) \cdot \alpha(b)\equiv \alpha(c)(\text{mod}\ 5)$

I'm stuck here... do I just plug in all the possible values of a, b, and c? I suppose there should be 4 ways to do this...

As an example:

$a = 1, b = 2, c = 3$

$1 + 2 = 3(\text{mod}\ 4)$

$3\equiv 3(\text{mod}\ 4)$

and

$\alpha(1) \cdot \alpha(2) \equiv \alpha(3)(\text{mod}\ 5)$

$2\cdot 3 \equiv 4(\text{mod}\ 5)$

$6\equiv 4(\text{mod}\ 5)??????$

I feel like I'm missing a key concept here. Been watching a ton of videos, but I'm missing something!