Lemma 14.2.4. The category $\Delta $ is the universal category with objects $[n]$, $n \geq 0$ and morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$ such that (a) every morphism is a composition of these morphisms, (b) the relations listed in Lemma 14.2.3 are satisfied, and (c) any relation among the morphisms is a consequence of those relations.

**Proof.**
Omitted.
$\square$

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