Uniqueness of Polynomial Ring in One Variable

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

Let $(R[X], \iota, X)$ and $(R[Y], \kappa, Y)$ be polynomial rings in one variable over $R$.

Then there exists a unique ring homomorphism $f : R[X] \to R[Y]$ such that:
 * $f \circ \iota = \kappa$
 * $f(X) = Y$

and it is an isomorphism.

Proof
The existence and uniqueness of $f$ follows from the universal property.

Likewise, there is a unique ring homomorphism $g : R[Y] \to R[X]$ such that:
 * $g \circ \kappa = \iota$
 * $g(Y) = X)$

and a unique ring homomorphism $h : R[X] \to R[X]$ such that:
 * $h \circ \iota = \iota$
 * $h(X) = X)$

By uniqueness and Identity Mapping is Ring Homomorphism, $h = \operatorname{id}$ is the identity mapping on $R[X]$.

Again by uniqueness and Composition of Ring Homomorphisms is Ring Homomorphism, $g\circ f = \operatorname{id}_{R[X]}$.

By symmetry, $f\circ g = \operatorname{id}_{R[Y]}$.

Thus $f$ is an isomorphism.