Let $\text{Ring}$ the category of commutative ring and $\text{Set}$ the category of set.
Denote by $\Omega : \text{Ring} \to \text{Set}$, the functor $R \mapsto \{ \text{Ideal of $R$} \}$, for a ring morphisme $\phi : R \to R'$, the induce map is $\Omega(\phi)(I) = \langle \phi(I) \rangle$ (the ideal of $R'$ generated by the image of $I$ by $\phi$.
Denote by $\mathbb{A}^n \setminus \{ 0 \}$ the functor define by $R \mapsto \{ x \in R^n \mid \langle x \rangle =R \}$, so the set of all $n$-tuple that generate the ideal $R$ of $R$.
I'm trying to descrybe all the natural transformation $\mathbb{A}^n \setminus \{ 0 \} \to \Omega$.
For an ideal $I$ of $\mathbb{Z}[x_1,\dots,x_n]$, i can create a natural transformation $\chi_I : \mathbb{A}^n \to \Omega$, where $\mathbb{A}^n$ is the functor $R \mapsto R^n$ because $\mathbb{A}^n$ is represented by $(\mathbb{Z}[x_1,\dots,x_n], (x_1,\dots,x_n))$ and the Yoneda lemma.
Next i can restriction $\chi_I$ to $\mathbb{A}^n \setminus \{ 0\}$. I have no idea if all natural transformation come in this way ? thx