Remark 1.4.6.13. Let $X$ be a Kan complex. By construction, the objects of the fundamental groupoid $\pi _{\leq 1}(X)$ are the vertices of $X$, and a pair of vertices $x,y \in X$ are isomorphic in $\pi _{\leq 1}( X )$ if and only if there exists an edge $e: x \rightarrow y$ in $X$. Applying Proposition 1.2.5.10, we deduce that $x,y \in X$ are isomorphic if and only if they belong to the same connected component of $X$. In other words, we have a canonical bijection
\[ \pi _0( X ) \simeq \{ \text{Objects of $\pi _{\leq 1}(X)$} \} / \text{Isomorphism}. \]