Lemma 14.5.2. Let $\mathcal{C}$ be a category.

1. Given a cosimplicial object $U$ in $\mathcal{C}$ we obtain a sequence of objects $U_ n = U([n])$ endowed with the morphisms $\delta ^ n_ j = U(\delta ^ n_ j) : U_{n - 1} \to U_ n$ and $\sigma ^ n_ j = U(\sigma ^ n_ j) : U_{n + 1} \to U_ n$. These morphisms satisfy the relations displayed in Lemma 14.2.3.

2. Conversely, given a sequence of objects $U_ n$ and morphisms $\delta ^ n_ j$, $\sigma ^ n_ j$ satisfying these relations there exists a unique cosimplicial object $U$ in $\mathcal{C}$ such that $U_ n = U([n])$, $\delta ^ n_ j = U(\delta ^ n_ j)$, and $\sigma ^ n_ j = U(\sigma ^ n_ j)$.

3. A morphism between cosimplicial objects $U$ and $U'$ is given by a family of morphisms $U_ n \to U'_ n$ commuting with the morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$.

Proof. This follows from Lemma 14.2.4. $\square$

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