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

Theorem
Let $L = \struct {S, \preceq}$ and $R = \struct {T, \precsim}$ be ordered sets.

Let $g: S \to T$ and $d: T \to S$ be mappings such that $\struct {g, d}$ is a Galois connection.

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

Sufficient Condition
Assume that
 * $d$ is a surjection.

By Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element
 * $\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$

By Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity:
 * $g \circ d = I_T$

Thus by Injection iff Left Inverse:
 * $d$ is an injection.

Necessary Condition
Assume that
 * $d$ is an injection.

By definition of Galois connection:
 * $g$ and $f$ are increasing mappings.

By Galois Connection Implies Order on Mappings:
 * $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$

By Increasing and Ordering on Mappings implies Mapping is Composition:
 * $d = \paren {d \circ g} \circ d$

By Composition of Mappings is Associative:
 * $= d \circ \paren {g \circ d}$

By Injection iff Left Cancellable:
 * $g \circ d = I_T$

By Identity Mapping is Right Identity:
 * $d = d \circ I_T$

Thus by Surjection iff Right Inverse:
 * $g$ is a surjection.