Remark 8.6.3.16. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let
\[ \lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]
be the projection maps of Notation 8.1.1.6. Then the morphism $\Xi $ of Construction 8.6.3.15 fits into a commutative diagram
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\begin{equation} \begin{gathered}\label{equation:compatibilities-of-Xi} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{ \rho _{-} } & \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \ar [l]_{\operatorname{ev}_0} \ar [r]^-{ \operatorname{ev}_{1} } \ar [d]^{\Xi } & \operatorname{\mathcal{C}}\ar [d]^{\rho _{+}} \\ \operatorname{Cospan}(\operatorname{\mathcal{C}}^{\operatorname{op}}) & \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [l]_{ \operatorname{Cospan}(\lambda _{-}) } \ar [r]^-{ \operatorname{Cospan}(\lambda _{+} ) } & \operatorname{Cospan}(\operatorname{\mathcal{C}}), } \end{gathered} \end{equation}
where $\rho _{+}$ and $\rho _{-}$ are the embeddings of Construction 8.1.7.1 and Variant 8.1.7.14. Moreover, the composition of $\Xi $ with the diagonal map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ is the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ of Construction 8.1.3.5.