Lemma 14.2.2. Any morphism in $\Delta$ can be written as a composition of the morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$.

Proof. Let $\varphi : [n] \to [m]$ be a morphism of $\Delta$. If $j \not\in \mathop{\mathrm{Im}}(\varphi )$, then we can write $\varphi$ as $\delta ^ m_ j \circ \psi$ for some morphism $\psi : [n] \to [m - 1]$. If $\varphi (j) = \varphi (j + 1)$ then we can write $\varphi$ as $\psi \circ \sigma ^{n - 1}_ j$ for some morphism $\psi : [n - 1] \to [m]$. The result follows because each replacement as above lowers $n + m$ and hence at some point $\varphi$ is both injective and surjective, hence an identity morphism. $\square$

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