Aluffi III.2.6 suggests proving the following:
Let $\alpha : R \rightarrow S$ be a fixed ring homomorphism, and let $s \in S$ be an element commuting with $\alpha(r)$ for all $r \in R$. Then there is a unique ring homomorphism $\overline \alpha : R[x] \rightarrow S$ extending $\alpha$ and sending $x$ to $s$.
I've managed to prove this by postulating the behaviour of $\overline \alpha$ on zero-degree polynomials (which coincide with $R$, thus $\overline \alpha$ coincides with $\alpha$ on those) and on $x$, then using the ring homomorphism properties to show that the whole $\overline \alpha$ is uniquely determined to prove its uniqueness, and then showing that $\overline \alpha$ indeed satisfies the homomorphism requirements (thus it's a well-defined homomorphism).
But this whole construction looks awfully lot like proving $\mathbb{Z}[x_1, \dots, x_n]$ satisfies the corresponding universal property in the corresponding category. Could I somehow reduce the original claim to the latter construction?
