Discrete Category is Order Category

Theorem
Let $\map {\mathbf{Dis} } S$ be a discrete category.

Then $\map {\mathbf{Dis} } S$ is also an order category.

Proof
We have, for any morphism $a \to b$ in $\map {\mathbf{Dis} } S$ that $a = b$.

Thus we see that $\map {\mathbf{Dis} } S$ will be an order category :


 * $\forall a, b \in S: a \preceq b \iff a = b$

holds for some ordering $\preceq$ on $S$.

The trivial ordering on $S$ accomplishes this.

Hence the result.