Ordered Set of All Mappings is Lattice iff Codomain is Lattice or Domain is Empty/Lemma

Lemma for Ordered Set of All Mappings is Lattice iff Codomain is Lattice or Domain is Empty
Let $S$ be a set.

Let $\struct {T, \preccurlyeq}$ be an ordered set.

Let $\struct {T^S, \preccurlyeq}$ denote the ordered set of all mappings from $S$ to $T$.

Let $S = \O$.

Then $\struct {T^S, \preccurlyeq}$ is a lattice.

Proof
Recall the definition of lattice:

Let $S = \O$.

Then there is one mapping from $S$ to $T$, and that is the empty mapping $e: S \to T$.

Thus we have $T^S = \set e$

From Supremum of Singleton and Infimum of Singleton:
 * $\sup \set e = e = \inf \set e$

Hence $T^S$ (trivially, in the degenerate sense) is a lattice.