Category Theory: Naturality and Notation I'm confused by some Category Theory notation but I give the whole question I'm interested in solving below for the sake of context.  Also, I want to verify my understanding of the proper approach to a solution assuming I can figure out this notation.
Here is the original question:

Let $\mathcal{F}:\mathbf{CRing}\rightarrow\mathbf{Set}$ be the
  forgetful functor mapping a commutative ring to its underlying set and
  mapping a ring homomorphism to itself. Show that $\mathcal{F}$ is
  represented by $\mathbf{Z}[T]$

1. The first part of my question is, what is $\mathbf{Z}[T]$ if $\mathbf{Z}$ is the additive group?
2. Now assuming I know what $\mathbf{Z}[T]$ is, in order to show that it represents $\mathcal{F}$ I just need to find a natural transformation from $\mathcal{F}$ to $\mathbf{Z}[T]$ such that an appropriate naturality square commutes, right?
 A: To show that the functor $\mathcal{F}$ is corepresented by $\mathbb{Z}[T]$, what you need in fact is a natural isomorphism from $\mathrm{Hom}(\mathbb{Z}[T],-)$ to $\mathcal{F}$.
For any category $\mathcal{C}$, any functor $\mathcal{F} : \mathcal{C} \rightarrow \mathrm{Set}$ and any element $X \in \mathcal{C}$, there is a bijective correspondence between the set of natural transformations $\mathrm{Mor}(X,-) \rightarrow \mathcal{F}$ and the set $\mathcal{F}(X)$: the element of $\mathcal{F}(X)$ associated to a natural transformation $\alpha : \mathrm{Mor}(X,-) \rightarrow \mathcal{F}$ is defined to be $\alpha_X(\mathrm{id}_X) \in \mathcal{F}(X)$; and conversely, if you have an element $u \in \mathcal{F}(X)$, the components $\alpha_Y : \mathrm{Mor}(X,Y) \rightarrow \mathcal{F}(Y)$ of the associated natural transformation $\alpha : \mathrm{Mor}(X,-) \rightarrow \mathcal{F}$ send a morphism $f : X \rightarrow Y$ to $\mathcal{F}(f)(u) \in \mathcal{F}(Y)$. This fact ( - and the analogous fact for contravariant functors $\mathcal{F} : \mathcal{C}^{\mathrm{op}} \rightarrow \mathrm{Set}$ - ) is known as the Yoneda lemma.
Now, in your situation, I claim that the natural transformation $\alpha : \mathrm{Hom}(\mathbb{Z}[T],-) \rightarrow \mathcal{F}$ corresponding to the element $T \in \mathcal{F}(\mathbb{Z}[T]) = \mathbb{Z}[T]$ is a natural isomorphism. To show that, we have to show that for every ring $R$, the function (of sets) $\alpha_R : \mathrm{Hom}(\mathbb{Z}[T],R) \rightarrow \mathcal{F}(R) = R$ is bijective. But for a ring homomorphism $f : \mathbb{Z}[T] \rightarrow R$, we have $\alpha_R(f) = \mathcal{F}(f)(T) = f(T)$, so that, indeed, $\alpha_R$ is bijective. (The set of ring homomorphisms $\mathbb{Z}[T] \rightarrow R$ corresponds bijectively to the set of elements of $R$, by associating to a ring homomorphism $f : \mathbb{Z}[T] \rightarrow R$ the element $f(T) \in R$.)
