Remark 8.1.1.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We will generally use Remark 8.1.1.2 to identify vertices of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with edges $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$. More generally, it will be useful to think of $n$-simplices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as encoding diagrams
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^{f_0} & X_1 \ar [l] \ar [d]^{f_1} & X_2 \ar [l] \ar [d]^{f_2} & \cdots \ar [l] \ar [d] & X_ n \ar [l] \ar [d]^{f_ n} \\ Y_0 \ar [r] & Y_1 \ar [r] & Y_2 \ar [r] & \cdots \ar [r] & Y_ n. } \]