I am reading Fraleigh and I am having trouble understanding the proof to this Lemma. It is my understanding that to prove that a map is an isomorphism, we must prove that it is a bijective homomorphism. I think the first half of this proof shows that the map is a homomorphism when the binary operation is addition and when the binary operation is multiplication. Then, Fraleigh has shown that the map is 1-1 but has now shown that the map is onto. Isn't this proof incomplete? What am I missing?
As Ihf notes, surjectivity is essentially "by choice", since Fraleigh restricts to $i[D]$.