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Remark Let $S_{\bullet }$ be a simplicial set. Proposition supplies a coequalizer diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ S_1 \ar@ <.4ex>[r]^-{d^{1}_0} \ar@ <-.4ex>[r]_-{d^{1}_1} & S_0 \ar [r] & \pi _0(S_{\bullet }).} \]

In other words, it allows us to identify $\pi _0( S_{\bullet } )$ with the quotient of $S_0 / \sim $, where $\sim $ is the equivalence relation generated by the set of edges of $S_{\bullet }$ (that is, the smallest equivalence relation with the property that $d^{1}_0(e) \sim d^{1}_1(e)$, for every edge $e \in S_1$). In particular, the set $\pi _0( S_{\bullet } )$ depends only on the $1$-skeleton of $S_{\bullet }$.