Galois Connection is Expressed by Minimum

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

Let $g: S \to T$, $d: T \to S$ be mappings.

Then $\left({g, d}\right)$ is Galois connection
 * $g$ is increasing mapping and
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

where
 * $\min$ denotes the minimum
 * $g^{-1}\left[{t^\succsim}\right]$ denotes the image of $t^\succsim$ under relation $g^{-1}$
 * $t^\succsim$ denotes the upper closure of $t$