Remark 2.4.6.2 (The Component Functor). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category (Construction 2.4.6.1). For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, Construction 1.2.1.18 supplies a map of simplicial sets
Here $\underline{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) }_{\bullet }$ denotes the constant simplicial set associated to the set $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$, and $u_{X,Y}$ carries each $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to the connected component which contains it. Allowing $X$ and $Y$ to vary, we obtain a simplicial functor $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ which is the identity on objects; we will refer to $u$ as the component functor.