# Discrete Category is Order Category

## Theorem

Let $\mathbf{Dis} \left({S}\right)$ be a discrete category.

Then $\mathbf{Dis} \left({S}\right)$ is also an order category.

## Proof

We have, for any morphism $a \to b$ in $\mathbf{Dis} \left({S}\right)$ that $a = b$.

Thus we see that $\mathbf{Dis} \left({S}\right)$ will be an order category iff:

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

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

The trivial ordering on $S$ accomplishes this.

Hence the result.

$\blacksquare$