Universal Property of Polynomial Ring/Free Monoid on Set

Theorem
Let $$R,S$$ be a commutative and unitary rings, and $$s_1,\ldots,s_n\in S$$.

Let $$\psi:R\to S$$ be a homomorphism of rings.

Let $$R[X_1,\ldots,X_n]$$ be the Ring of Polynomial Forms in $$n$$ indeterminates with coefficients in $$R$$.

Then there exists a unique homomorphism $$\phi:R[X_1,\ldots,X_n]\to S$$ extending $$\psi$$ such that $$\phi(X_j)=s_j$$ for $$j=1,\ldots,n$$.

Remarks

 * The homomorphism $$\phi$$ is often called "evaluation at s".


 * The requirement that the rings be commutative is vital. A fundamental difference for polynomials over non-commutative rings is additional difficulty identifying polynomial forms and functions using this method.