Example 10.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. The composite functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
is a semisimplicial object of $\operatorname{\mathcal{C}}$, which we will refer to as the underlying semisimplicial object of $X_{\bullet }$. We will often abuse notation by identifying $X_{\bullet }$ with its underlying semisimplicial object. Similarly, every cosimplicial object $X^{\bullet }$ of $\operatorname{\mathcal{C}}$ has an underlying cosemisimplicial object, given by the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \xrightarrow { X^{\bullet } } \operatorname{\mathcal{C}}. \]