Kerodon

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Corollary 1.1.4.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $C \mapsto \underline{C}_{\bullet }$ 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 $\operatorname{\mathcal{C}}$.

Proof. Let $C$ and $D$ be objects of $\operatorname{\mathcal{C}}$; we wish to show that the canonical map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, \underline{\operatorname{\mathcal{D}}}_{\bullet } ) \]

is a bijection. This is clear, since $\theta $ is right inverse to the evaluation map

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, \underline{\operatorname{\mathcal{D}}}_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \]

which is bijective by virtue of Proposition 1.1.4.5. $\square$