Kerodon

\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit2} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{D}})}( \overline{F}|_{\operatorname{\mathcal{C}}^1} , \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] \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 [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{E}})}( U \circ \overline{F}|_{\operatorname{\mathcal{C}}^1}, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^0}) } \end{gathered} \end{equation}