All Infima Preserving Mapping is Upper Adjoint of Galois Connection

Theorem
Let $\left({S, \preceq}\right)$ be a complete lattice.

Let $\left({T, \precsim}\right)$ be an ordered set.

Let $g: S \to T$ be all infima preserving mapping.

Then there exists a mapping $d: T \to S$ such that $\left({g, d}\right)$ is Galois connection 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 $t$

Proof
Define a mapping $d: T \to S$:
 * $\forall t \in T:d\left({t}\right) := \inf\left({g^{-1}\left[{t^\succsim}\right]}\right)$

We will prove as lemma 1 that
 * $g$ is an increasing mapping.

Let $x, y \in S$ such that
 * $x \preceq y$

By Upper Closure is Decreasing:
 * $y^\succeq \subseteq x^\succeq$

By Infimum of Upper Closure of Element:
 * $\inf\left({x^\succeq}\right) = x$ and $\inf\left({y^\succeq}\right) = y$

By definition of mapping preserves all infima:
 * $g$ preserves the infimum on $x^\succeq$ and $g$ preserves the infimum on $y^\succeq$

By definition of mapping preserves the infimum of set:
 * $\inf\left({g^\to\left({x^\succeq}\right)}\right) = g\left({x}\right)$ and $\inf\left({g^\to\left({y^\succeq}\right)}\right) = g\left({y}\right)$

By Image of Subset under Relation is Subset of Image/Corollary 2:
 * $g^\to\left({y^\succeq}\right) \subseteq g^\to\left({x^\succeq}\right)$

Thus by Infimum of Subset:
 * $g\left({x}\right) \precsim g\left({y}\right)$

This ends the proof of lemma 1.

We will prove as lemma 2 that
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

Let $t \in T$.

By definition of $d$:
 * $d\left({t}\right) = \inf\left({g^{-1}\left[{t^\succsim}\right]}\right)$

By Image of Inverse Image:
 * $g\left[{g^{-1}\left[{t^\succsim}\right]}\right] \subseteq t^\succsim$

By Infimum of Subset and Infimum of Upper Closure of Element:
 * $t = \inf\left({t^\succsim}\right) \precsim \inf\left({g\left[{g^{-1}\left[{t^\succsim}\right]}\right]}\right)$

By definition of upper closure of element:
 * $\inf\left({g\left[{g^{-1}\left[{t^\succsim}\right]}\right]}\right) \in t^\succsim$

By definition of complete lattice:
 * $g^{-1}\left[{t^\succsim}\right]$ admits an infimum

By definitions mapping preserves the infimum:
 * $\inf\left({g\left[{g^{-1}\left[{t^\succsim}\right]}\right]}\right) = g\left({d\left({t}\right)}\right)$

Thus by definition of image of set:
 * $d\left({t}\right) \in g^{-1}\left[{t^\succsim}\right]$

Thus
 * $d\left({t}\right) = \min \left({ g^{-1}\left[{t^\succsim}\right] }\right)$

This ends the proof of lemma 2.

Thus by Galois Connection is Expressed by Minimum:
 * $\left({g, d}\right)$ is a Galois connection.

Thus by lemma 2:
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$