Find all the homomorphisms from Z -> Z x Z and Z x Z -> Z.

For the first, I've found and proven the following.

Z -> Z x Z such that a -> (a,0) or (0,a) or (0,0).

For the second one:

Z x Z -> Z such that (a,b)-> a or b (the "forgetful" function) or 0.

So I have found three. Am I on the right track? I have verified that these are indeed a homomorphism.