Kerodon

\begin{equation} \begin{gathered}\label{equation:diagram-Kan-slice-compatibility} \xymatrix@C =-10pt@R=40pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \ar [rr] \ar [dr]_{\rho \circ \overline{\theta } } \ar [dd] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [dr]_{\lambda } \ar [dd]^(.6){\circ f} & \\ & \operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}}^{\mathrm{L}}( (Y,f), (Z,g) ) \ar [rr] \ar [dd] & & Q \ar [dd]^{\pi '} \\ \{ g\} \ar@ {=}[dr] \ar [rr] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [dr]_-{\theta _{X,Z}} & \\ & \{ g\} \ar [rr] & & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) } \end{gathered} \end{equation}