A natural transformation between two specific categories

Question

I was looking at the following example of a natural transformation from Hilton-Stambach's homological algebra.

I didn't understand about $\tau$; it is natural transformation between what functors? Can one elaborate this example a little?

Answer 1

$F$ is the functor that associates to a set $X$ the free abelian group $\mathbb{Z} \left<X\right>$ on the elements of $X$ (in that book's notation, $\mathbb{Z} \left<X\right> = X_F$). The functor $U\colon \mathbf{Ab} \to \mathbf{Set}$ is also known as the forgetful functor (to every abelian group, it associates the underlying set). By $I$ they denote the identity functor $\mathrm{Id}_\mathbf{Ab}$ on the category of abelian groups.

Let me explain where this natural transformation comes from (because it arises really naturally). The universal property of free abelian groups boils down to a natural bijection (adjunction of functors) $$\operatorname{Hom}_\mathbf{Ab} (\mathbb{Z} \left<X\right>, A) \cong \operatorname{Hom}_\mathbf{Set} (X, U (A))$$ In words: a homomorphism from a free group $\mathbb{Z} \left<X\right>$ to any group $A$ is determined by the images of the generators.

This natural transformation $\tau\colon F\circ U \Rightarrow \mathrm{Id}_\mathbf{Ab}$ between the composition of $U$ with $F$ and the identity functor on $\mathbf{Ab}$ is the adjunction counit. The morphism $\tau_A\colon F U (A) \to A$ is the morphism that corresponds under the bijection above to the identity morphism $U (A) \to U (A)$.

We have also the adjunction unit, which is a natural transformation $\eta\colon \mathrm{Id}_\mathbf{Set} \Rightarrow U\circ F$; the map between sets $\eta_X\colon X\to U F (X)$ is the arrow that corresponds to the identity $F (X) \to F (X)$. It is the inclusion of $X$ as generators of $\mathbb{Z} \left<X\right>$.

I guess the authors come back to this example in the context of the adjunction between $F$ and $U$.