It is well known endomorphisms of faithful functor form a monoid. I was trying to determine monoid of endomorphisms of forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, and found it to be multiplicative monoid of $\mathbb{Z}$. Proof goes like this:
Let $U$ be the forgetful functor, and $\theta\colon U \to U$ a natural transformation. Consider infinite cyclic group $A=\left<a\right>$. We have $\theta_A(a) = a^k$ for some integer $k$. Now take arbitrary group $G$ and $g\in G$, and consider homomorphism $f\colon A\to G$ determined by $f(a) = g$. Clearly, $$ \theta_G\left(g\right) = \theta_G\left(f\left(a\right)\right) = \left(\theta_G \circ f\right)\left(a\right) = \left(f\circ \theta_A\right)\left(a\right) = f\left(\theta_A(a)\right) = f(a^k)=\left(f(a)\right)^k = g^k $$ the only nontrivial equality true by naturality of $\theta$, and so endomorphisms of $U$ correspond to integers, composition given by multiplication.
Now, for the questions...
Is the above correct? It surely seems to be, but I find it kind of unsettling, especially since...
... the same reasoning seems to apply to $\mathbf{Ab}\to \mathbf{Set}$. Is the same indeed true for $\mathbf{Ab}$? I'd intuitively expect richer structure in the nonabelian case.
The role played by the cyclic group ($\mathbb{Z}$) seems worth further consideration. I know $\mathbb{Z}$ is a generator (separator) in $\mathbf{Grp}$, but it doesn't seem to immediately lead to any useful generalizations. Is there some connection between endomorphisms of forgetful functor and separators? Or some other related notion?
What about some "more abstract" ways to answer such questions? That is, instead of element chasing perhaps some variant of Yoneda lemma for suitably enriched categories?