Proposition 10.2.2.25. Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. The following data are equivalent:
Extensions of $X_{\bullet }$ to an augmented semisimplicial object $\overline{X}_{\bullet }$ satisfying $\overline{X}_{-1} = X$.
Morphisms $\epsilon : X_0 \rightarrow X$ satisfying $\epsilon \circ d^{1}_0 = \epsilon \circ d^{1}_1$, where $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$ are the face operators of the semisimplicial object $X_{\bullet }$.
Here the equivalence is implemented by taking $\epsilon $ to be the face operator $d^{0}_{0}: X_{0} \rightarrow X_{-1}$ of Remark 10.2.2.19.