Discrete Category on Set is Discrete Category

Theorem
Let $S$ be a set.

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

Then $\mathbf{Dis} \left({S}\right)$ determines a unique (up to isomorphism discrete category $\mathbf{Dis} \left({S}\right)$ whose objects precisely comprise $S$.