Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 1.1.6.19. 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 1.1.6.18. 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 1.1.6.12. $\square$