# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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. $\square$