This is some sort of follow up question to https://math.stackexchange.com/q/141901.
I understand that to show the normality of the commutator subgroup it is enough to show that $g[a,b]g^{-1}$ is in the commutator subgroup.
In the above answer and in many articles I read, comes the homomorphism part in the proof. I have been processing it for hours but I really don't understand how we draw the conclusion.
As far as I understand it, we define the homomorphism as $f(x)=gxg^{-1}$ and use the fact that $f([x,y])=[f(x),f(y)]$. Why and how do we use this, and why the equality holds true and how do arrive to the desired conclusion.
I would be really thankful for help. Clearly I am missing something or have problems with basic idea of homomorphism.
Thanks in advance