Kerodon

\begin{equation} \begin{gathered}\label{equation:composition-law-on-dg-nerve} \xymatrix { \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \ar [r]^-{\mu } \ar [d]^{ \rho _{Z,Y} \times \rho _{Y,X} } & \mathrm{K}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }) \ar [d]^{ \rho _{Z,X} } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }(Y,Z) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X,Y) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z) } \end{gathered} \end{equation}