$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
7.1
\begin{equation} \begin{gathered}\label{equation:U-colimit-by-morphism-space} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \underline{C}, \overline{f} ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \underline{C}|_{K}, \overline{f}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{D}}) }( U \circ \underline{C}, U \circ \overline{f} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ \underline{C}|_{K}, U \circ \overline{f}|_{K} ) } \end{gathered} \end{equation}