$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.56
\begin{equation} \begin{gathered}\label{equation:witness-to-strict-Hom} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \ar [r]^-{ \widetilde{\mathscr {H}}_{\operatorname{\mathcal{C}}} } \ar [d] & \operatorname{\mathcal{S}}_{\ast } \ar [d]^{U} \\ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \ar [r]^-{ \mathscr {H}_{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{S}}, } \end{gathered} \end{equation}