I don't understand it. See for it to be isomorphism you need to it to be homomorphism between them. I can see how this is trivial from what's worked out. I can sort of agree that surjection condition is obvious. However, don't see how he proves injection.
If $af=bf$, then I see that $a-b \in Kerf$. However, I don't see how this gives the injection condition. Looking at the proofwiki doesn't help as it uses a totally different proof. Plus can't get algebra 2 notes that I used last year.