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Remark The category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ of Notation can be regarded as a (non-full) subcategory of the simplex category $\operatorname{{\bf \Delta }}$ of Definition Consequently, any simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ has an underlying semisimplicial object, given by the composition

\[ \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}. \]

We will often abuse notation by identifying a simplicial object of $\operatorname{\mathcal{C}}$ with its underlying semisimplicial object.