Lemma 14.20.2. Let $\mathcal{C}$ be a category. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $U$ be a simplicial object of $\mathcal{C}$. To give an augmentation of $U$ towards $X$ is the same as giving a morphism $\epsilon _0 : U_0 \to X$ such that $\epsilon _0 \circ d^1_0 = \epsilon _0 \circ d^1_1$.
Proof. Given a morphism $\epsilon : U \to X$ we certainly obtain an $\epsilon _0$ as in the lemma. Conversely, given $\epsilon _0$ as in the lemma, define $\epsilon _ n : U_ n \to X$ by choosing any morphism $\alpha : [0] \to [n]$ and taking $\epsilon _ n = \epsilon _0 \circ U(\alpha )$. Namely, if $\beta : [0] \to [n]$ is another choice, then there exists a morphism $\gamma : [1] \to [n]$ such that $\alpha $ and $\beta $ both factor as $[0] \to [1] \to [n]$. Hence the condition on $\epsilon _0$ shows that $\epsilon _ n$ is well defined. Then it is easy to show that $(\epsilon _ n) : U \to X$ is a morphism of simplicial objects. $\square$
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Comment #1018 by correction_bot on