Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 1.2.1.19. Let $S_{}$ be a simplicial set and let $u: S_{} \rightarrow \underline{ \pi _0( S_{})}_{}$ be the component map of Construction 1.2.1.18. For every set $J$, composition with $u$ induces a bijection

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

Proof. Decomposing $S_{}$ as the union of its connected components, we can reduce to the case where $S_{}$ is connected, in which case the desired result is a reformulation of Corollary 1.2.1.12. $\square$