Kerodon

\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})}( \overline{F}, \underline{D} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{E}})}( U \circ \overline{F}, U \circ \underline{D} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, \underline{D}|_{ \operatorname{\mathcal{C}}} ) \ar [d] \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}}) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})}( F|_{ \operatorname{\mathcal{C}}^{0} }, \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^0, \operatorname{\mathcal{E}})}( U \circ F|_{ \operatorname{\mathcal{C}}^{0} }, U \circ \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } )} \end{gathered} \end{equation}