Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable

Theorem
Let $R$ be a commutative ring with unity.

Let $R[X]$ be a polynomial ring in one variable $X$ over $R$.

Let $\iota : R \to R[X]$ denote the canonical embedding.

Let $S$ be a commutative ring with unity and $f : R[X] \to S$ be a ring isomorphism.

Then $(S, f \circ \iota, f(X))$ is a polynomial ring in one variable $f(X)$ over $R$.