Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection

Theorem
Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be ordered sets.

Ley $g:S \to T, d:T \to S$ be mappings such that
 * $\left({g, d}\right)$ is Galois connection.

Then $g$ is a surjection $d$ is an injection.