Discrete Category is Order Category

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

Then $\mathbf{Dis} \left({S}\right)$ is also a poset 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 a poset 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.