Remark 11.9.1.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the morphism $\Xi $ of Construction 11.9.1.3 can be described concretely on low-dimensional simplices as follows:
On vertices, $\Xi $ is given by the formula $\Xi (f) = f$. Here we abuse notation by identifying vertices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$ with edges of the simplicial set $\operatorname{\mathcal{C}}$.
Let $e: f_0 \rightarrow f_1$ be an edge of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we identify with a $3$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ displayed informally in the diagram
\[ \xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{ f_0 } & X_1 \ar [l]_{g} \ar [d]^{f_1} \\ Y_0 \ar [r]^-{h} & Y_1. } \]Then $\Xi (e)$ is the cospan from $f_0$ to $f_1$ in the simplicial set $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ depicted informally in the diagram
\[ \xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{f_0} \ar [r]^-{\operatorname{id}} & X_0 \ar [d] & X_1 \ar [d]^{f_1} \ar [l]_{g} \\ Y_0 \ar [r]^-{h} & Y_1 & Y_1. \ar [l]_{\operatorname{id}} } \]