$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
5.10
\begin{equation} \begin{gathered}\label{equation:fully-faithful-check-cartesian2} \xymatrix@R =50pt@C=25pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X',Y',Z') \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y',Z') } \{ g'\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X',Z') \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( U'(X'), U'(Y'), U'(Z') ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( U'(Y), U'(Z)) } \{ U'(g') \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( U'(X'), U'(Z') ). } \end{gathered} \end{equation}