Corollary 2.5.6.13. The normalized chain complex functor
\[ \mathrm{N}_{\ast }( -; \operatorname{\mathbf{Z}}): \operatorname{Set_{\Delta }}\rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}}) \]
admits a right adjoint, given on objects by the functor $M_{\ast } \mapsto \mathrm{K}( M_{\ast } )$ of Construction 2.5.6.3.