Corollary 1.2.1.21. The connected component functor
\[ \pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad S_{} \mapsto \pi _0( S_{} ) \]
of Remark 1.2.1.17 is left adjoint to the constant simplicial set functor
\[ \operatorname{Set}\rightarrow \operatorname{Set_{\Delta }}\quad \quad I \mapsto \underline{I}_{} \]
of Construction 1.1.5.2. More precisely, the construction $S_{} \mapsto (u: S_{} \rightarrow \underline{ \pi _0( S_{})}_{})$ is the unit of an adjunction.