Dedekind Completion is Unique up to Isomorphism

Theorem
Let $S$ be an ordered set.

Let $\left({X, f}\right)$ and $\left({Y, g}\right)$ be Dedekind completions of $S$.

Then there exists a unique order isomorphism $\psi: X \to Y$ such that $\psi \circ f = g$.

Also see

 * Existence of Dedekind Completion