Proving commutator subgroup is normal

Question

Part of Aluffi II.7.11 suggests proving this claim.

It is actually quite straightforward (especially after some earlier problems in the book): I've managed to prove it by noticing that for arbitrary $a, b, g$: $$g aba^{-1} b^{-1} g^{-1} = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}$$ which is in $[G, G]$, thus it's trivial to show that for arbitrary $n \in [G, G]$ the element $gng^{-1}$ is also in $[G, G]$ by interspersing it with some number of $g^{-1}g$'s.

My question, though, is a bit different (and that's what's different from all other questions about normality of commutators that MSE suggests as similar). Aluffi gives a hint that $g \cdot aba^{-1}b^{-1} \cdot g^{-1} = \gamma_g(aba^{-1}b^{-1})$, where $\gamma_g : a \mapsto gag^{-1}$ is an inner automorphism on $G$ defined by the element $g$. I don't think I've followed this hint and thus I haven't built a "more categorical" proof that Aluffi might have expected here. So, is there a way to prove the claim using this hint?

Answer 1

You implicitly did follow the hint:

The point is that $\gamma_g$ is an inner automorphism, implying it's a group homomorphism, so that $$\gamma_g(xy) =\gamma_g(x)\gamma_g(y)$$ which was the crucial part in your proof, as well.

Answer 2

I think @Andreas Blass in his comment has the key to the "more categorical" approach.

The commutator can be viewed as a functor on the category of groups. The Wikipedia page is good on this.

It follows that the commutator subgroup is a fully characteristic subgroup,(ie, it is stable under any endomorphism).

Thus it is automatically a normal subgroup. This is because the normality condition can be phrased as invariance under inner automorphisms.