Kerodon

\begin{equation} \begin{gathered}\label{equation:enriched-homotopy-transport-universal} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{\mathcal{C}}'}(Y,Z)_{\bullet } \ar [r] \ar [d]^-{\theta _{Y,Z}^{\simeq } } & \operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet })^{\simeq } \ar [dd]^{ \theta _{X,Z} \circ } \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')}(Y,Z) \ar [d]^-{\rho } & \\ \operatorname{Fun}( \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Y), \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Z) )^{\simeq } \ar [r]^-{ \circ \theta _{X,Y} } & \operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }^{\mathrm{L}}(X,Z) )^{\simeq } } \end{gathered} \end{equation}