Corollary 1.1.5.9. Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $C \mapsto \underline{C}_{}$ determines a fully faithful embedding from $\operatorname{\mathcal{C}}$ to the category $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ of simplicial objects of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $C$ and $D$ be objects of $\operatorname{\mathcal{C}}$; we wish to show that the canonical map
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{}, \underline{D}) \]
is a bijection. This is clear, since $\theta $ is right inverse to the evaluation map
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{}, \underline{D} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \]
which is bijective by virtue of Proposition 1.1.5.5. $\square$