# Kerodon

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Remark 2.4.2.6. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{\mathcal{D}}_{\bullet }$ be a simplicial category. Applying Proposition 1.1.4.5 (and Remark 2.4.1.12), we deduce that the restriction map

$\{ \text{Simplicial functors \underline{\operatorname{\mathcal{C}}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }} \} \simeq \{ \text{Functors \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_0} \} ,$

is bijective. In other words, the fully faithful embedding

$\operatorname{Cat}\hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{\mathcal{C}}}_{\bullet }$

of Remark 2.4.2.5 is left adjoint to the forgetful functor

$\operatorname{Cat_{\Delta }}\rightarrow \operatorname{Cat}\quad \quad \operatorname{\mathcal{D}}_{\bullet } \mapsto \operatorname{\mathcal{D}}_0.$

of Example 2.4.1.4.