Remark 10.2.2.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. By virtue of Remark 10.2.1.11, the following data are equivalent:
Augmented semisimplicial objects $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carrying the object $[-1]$ to $X$.
Semisimplicial objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.
We will often invoke this equivalence implicitly, using the notation $X_{\bullet }$ to indicate both an augmented simplicial object of $\operatorname{\mathcal{C}}$ (satisfying $X_{-1} = X$) and the associated simplicial object of $\operatorname{\mathcal{C}}_{/X}$.