Ordered Semigroup Monomorphism into Image is Isomorphism

Theorem
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups.

Let $\phi: \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ be an ordered semigroup monomorphism.

Let $S\,'$ be the image of $\phi$.

Then $\phi$ is an ordered semigroup isomorphism from $\left({S, \circ, \preceq}\right)$ into $\left({S\,', * \restriction_{S\,'}, \preccurlyeq \restriction_{S\,'}}\right)$.

Here:
 * $* \restriction_{S\,'}$ denotes the restriction of $*$ to $S\,' \times S\,'$
 * $\preccurlyeq \restriction_{S\,'}$ denotes the restriction of $\preccurlyeq$ to $S\,' \times S\,'$.

Proof
Let $\phi: \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ be an ordered semigroup monomorphism.

Then $\phi$ is an injection into $\left({T, *, \preccurlyeq}\right)$ by definition.

From Surjection iff Image equals Codomain, any mapping from a set to the image of that mapping is a surjection.

Thus the surjective restriction of $\phi$ onto $S\,'$ is an ordered semigroup monomorphism which is also a surjection.

Hence the result from Ordered Semigroup Isomorphism is Surjective Monomorphism.