Natural transformations $Id_{Ab} \rightarrow Id_{Ab}$

Question

Let $Id:\frak{Ab} \rightarrow {Ab}$ be the identity functor of $\frak{Ab}$ (category of abelian groups). The class of natural transformations $\eta: Id \rightarrow Id$ is a monoid under operation defined as follows: $$\eta \circ \varepsilon = \{\eta_G\}_{G \in \frak{Ab}} \circ \{\varepsilon_G\}_{G \in \frak{Ab}} := \{\eta_G \circ \varepsilon_G\}_{G \in \frak{Ab}}$$ The unit of that monoid is the identity transformation $id := \{id_G\}_{G \in \frak{Ab}}$.

The task is to calculate that monoid.

I am able to translate the problem to group theory. I believe that for any abelian group $G$ I must determine all homomorphisms $\alpha_G: G\rightarrow G$, such that $\phi \circ \alpha_G = \alpha_H \circ \phi$ holds for any abelian groups $G, H$ and any homomorphism $\phi: G\rightarrow H$.

My attempts got no further than failing at providing examples... I guess group automorphisms don't work since we can fix $\beta: x \mapsto -x$ for $\mathbb Z$ and $\gamma: x \mapsto x$ for $\mathbb Z_3$, and then for $\alpha: \mathbb Z \rightarrow \mathbb Z_3, x \mapsto x \text{ mod } 3$ the condition $\gamma\circ\alpha=\alpha\circ\beta$ will fail.

Answer 1

$\text{End}(\text{Id})$ is actually a commutative ring, not just a monoid. The commutativity is by the Eckmann-Hilton argument and the addition is pointwise.

I believe that for any abelian group $G$ I must determine all homomorphisms

You're getting a little mixed up about the quantifiers here, I think. The question is to determine all collections of homomorphisms $\alpha_G : G \to G$ which are natural in $G$. It's not to figure something out one group at a time; you need $\alpha_G$ for every $G$ simultaneously. The simplest non-identity examples are to take every $\alpha_G$ to be zero $x \mapsto 0$, or to be inversion $x \mapsto -x$ (check that these are natural).

This is a priori quite a lot of data so you might be worried that there's a huge variety of different possible choices but actually naturality is a very strong constraint. Here's a sequence of hints / exercises.

Exercise / Hint 1: Prove that a natural transformation $\alpha$ as above is determined by $\alpha_{\mathbb{Z}} : \mathbb{Z} \to \mathbb{Z}$.

Exercise / Hint 2: Prove that every possible choice of $\alpha_{\mathbb{Z}} : \mathbb{Z} \to \mathbb{Z}$ extends uniquely to a natural transformation $\alpha$ as above. Conclude that $\text{End}(\text{Id}) \cong \text{End}(\mathbb{Z}) \cong \mathbb{Z}$ (as a commutative ring, not just as a monoid).

Exercise 3: Generalize this argument replacing $\text{Ab}$ with the category $\text{Mod}(R)$ of modules over a ring, not necessarily commutative. In this case it's again true that $\text{End}(\text{Id})$ has the structure of a commutative ring. What commutative ring do you get?