Kerodon

Remark 2.5.7.4 (Unitality of the Shuffle Product). Let $\operatorname{\mathbf{Z}}[ \Delta ^{0} ]$ be the constant simplicial abelian group taking the value $\operatorname{\mathbf{Z}}$, and let us identify the integer $1$ with the corresponding $0$-simplex of $\operatorname{\mathbf{Z}}[ \Delta ^{0} ]$. Then, for any simplicial abelian group $A_{\bullet }$, the canonical isomorphisms $A_{\bullet } \simeq (A \otimes \operatorname{\mathbf{Z}}[\Delta ^{0}])_{\bullet }$ and $A_{\bullet } \simeq (\operatorname{\mathbf{Z}}[ \Delta ^{0} ] \otimes A )_{\bullet }$ are given by $a \mapsto a \bar{\triangledown } 1$ and $a \mapsto 1 \bar{\triangledown } a$, respectively.