# Kerodon

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Remark 3.1.5.12. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category (Definition 2.4.1.8). Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of Construction 2.4.6.1 inherits the structure of an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category, which can be described concretely as follows:

• For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the mapping object $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Y )$ is the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, regarded as an object of $\mathrm{h} \mathit{\operatorname{Kan}}$.

• For every pair of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition law

$\underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( Y,Z ) \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Y ) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Z )$

is the homotopy class of the composition map $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$.