$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Construction 8.6.3.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and suppose we are given a pair of morphisms $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. Let $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets for which the diagram
8.76
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}
is commutative. Let $\lambda _{+}: \operatorname{Tw}( \operatorname{\mathcal{E}}^{\dagger } ) \rightarrow \operatorname{\mathcal{E}}^{\dagger }$ be the projection map of Notation 8.1.1.6 and let $\iota : \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}}) \xrightarrow {\sim } \operatorname{Tw}(\operatorname{\mathcal{C}})$ be the isomorphism described in Remark 8.1.1.7. Then we can extend (8.76) to a commutative diagram
8.77
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy2} \xymatrix@C =50pt@R=50pt{ \operatorname{Tw}(\operatorname{\mathcal{E}}^{\dagger }) \ar [r]^-{(\lambda _{+}, \iota \circ \operatorname{Tw}(U^{\dagger }))} \ar [d]^{ \operatorname{Tw}(U^{\dagger }) } & \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} ) \ar [r]^-{ \iota } & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Using Proposition 8.1.3.7, we can identify the outer rectangle with a diagram
8.78
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy3} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \ar [r] \ar [d]^{U^{\dagger }} & \operatorname{Cospan}( \operatorname{\mathcal{E}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [r]^-{\rho _{-}} & \operatorname{Cospan}( \operatorname{\mathcal{C}}), } \end{gathered} \end{equation}
which we can identify with a comparison map $\Psi : \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}^{\operatorname{op}}$.