# Discrete Category on Set is Discrete Category

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## 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 $S$.

## Proof

$\blacksquare$

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4.12$