Example 1.1.1.8. Let $S_{\bullet }$ be a simplicial set and let $\sigma $ be a $2$-simplex of $S_{\bullet }$. Then $\sigma $ has three faces: the edges $f = d^{2}_{2}(\sigma )$, $g = d^{2}_{0}(\sigma )$, and $h = d^{2}_{1}(\sigma )$. In this case, Remark 1.1.1.7 asserts the following:
The edges $f$ and $h$ have the same source vertex $x \in S_{\bullet }$.
The edges $g$ and $h$ have the same target vertex $z \in S_{\bullet }$.
The target of $f$ and the source of $g$ are the same vertex $y \in S_{\bullet }$.
These relationships can be encoded visually in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]