# Category of Ordered Sets is Category

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## Theorem

Let $\mathbf{OrdSet}$ be the category of ordered sets.

Then $\mathbf{OrdSet}$ is a metacategory.

## Proof

Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.

For any two increasing mappings their composition (in the usual set theoretic sense) is again increasing by Composite of Increasing Mappings is Increasing.

For any set $X$, we have the identity mapping $\operatorname{id}_X$.

By Identity Mapping is Left Identity and Identity Mapping is Right Identity that this is the identity morphism for $X$.

That it is increasing follows from Identity Mapping is Increasing.

Finally by Composition of Mappings is Associative, the associative property is satisfied.

Hence $\mathbf{OrdSet}$ is a metacategory.

$\blacksquare$

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4.3$