Order Isomorphism between Tosets is not necessarily Unique

Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be tosets.

Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \preccurlyeq_2}$, that is, let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be order isomorphic.

Then it is not necessarily the case that there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.

Proof
Proof by Counterexample:

Let $\Z$ denote the set of integers.

We have that Integers under Usual Ordering form Totally Ordered Set.

Let $m \in \Z$ be an arbitrary integer.

Let $f_m: \Z \to \Z$ be the order isomorphism defined as:


 * $\forall a \in \Z: \map {f_m} a = a + m$

It follows that there are at least as many order isomorphisms of from $\Z$ to $\Z$ as there are integers.

Hence the result