Lemma 14.5.7. Let $\mathcal{C}$ be a category. Let $U$ be a cosimplicial object of $\mathcal{C}$. Each of the morphisms $\delta ^ n_ i : U_{n - 1} \to U_ n$ has a left inverse. In particular $\delta ^ n_ i$ is a monomorphism.

**Proof.**
This is true because $\sigma _ i^{n - 1} \circ \delta ^ n_ i = \text{id}_{U_ n}$ for $j < n$.
$\square$

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