Galois Connection Implies Order on Mappings

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

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

Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$

where
 * $\preceq, \precsim$ denote the orderings on mappings,
 * $I_S$ denotes the identity mapping of $S$

Proof
Let $s \in S$.

By definition of reflexivity:
 * $\map g s \precsim \map g s$

By definition of Galois connection:
 * $\map d {\map g s} \preceq s$

By definition of composition:
 * $\map {\paren {d \circ g} } s \preceq s$

By definition of identity mapping:
 * $\map {\paren {d \circ g} } s \preceq \map {I_S} s$

Thus by definition of order on mappings:
 * $d \circ g \preceq I_S$

Let $t \in T$.

By definition of reflexivity:
 * $\map d t \preceq \map d t$

By definition of Galois connection:
 * $t \precsim \map g {\map d t}$

By definition of composition:
 * $t \precsim \map {\paren {g \circ d} } t$

By definition of identity mapping:
 * $\map {I_T} t \precsim \map {\paren {g \circ d} } t$

Thus by definition of order on mappings:
 * $I_T \precsim g \circ d$