Kerodon

Remark 1.2.1.17 (Functoriality of $\pi _0$). Let $f: S_{} \rightarrow T_{}$ be a map of simplicial sets. It follows from Proposition 1.2.1.11 that for each connected component $S'_{} \subseteq S_{}$, there is a unique connected component $T'_{} \subseteq T_{}$ such that $f(S'_{} ) \subseteq T'_{}$. The construction $S'_{} \mapsto T'_{}$ then determines a map of sets $\pi _0(f): \pi _0(S_{}) \rightarrow \pi _{0}( T_{} )$. This construction is compatible with composition, and therefore allows us to view the construction $S_{} \mapsto \pi _0( S_{} )$ as a functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ from the category of simplicial sets to the category of sets.