Ordered Fields in Which Every Nonempty Set Bounded Above Has a Supremum Are Unique up to Unique Isomorphism

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $F$ and $G$ be ordered fields in each of which every nonempty subset of it that is bounded above has a supremum, with operations and orders written alike, as in Homomorphisms and Isomorphisms of Ordered Fields, and let $\kappa_{F}:\mathbb{Q}\to F$ and $\kappa_{G}:\mathbb{Q}\to G$ be the canonical embeddings.

¶ There is exactly one homomorphism of ordered fields $f:F\to G$.

¶ This $f$ is an isomorphism of ordered fields.

¶ $f\circ\kappa_{F}=\kappa_{G}$.