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Proposition Let $S_{\bullet }$ be a simplicial set and let $u: S_{\bullet } \rightarrow \underline{ \pi _0( S_{\bullet })}_{\bullet }$ be the component map of Construction For every set $J$, composition with $u$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Set}}( \pi _0(S_{\bullet }), J ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \underline{J}_{\bullet } ). \]

Proof. Decomposing $S_{\bullet }$ as the union of its connected components, we can reduce to the case where $S_{\bullet }$ is connected, in which case the desired result is a reformulation of Corollary $\square$