Kerodon

Remark 2.5.6.20. Let $A_{\bullet }$ be a simplicial abelian group. Then Warning 2.5.6.19 supplies a canonical isomorphism of normalized Moore complexes $\mathrm{N}_{\ast }(A) \simeq \mathrm{N}_{\ast }( A^{\operatorname{op}} )$. By virtue of Theorem 2.5.6.1, this isomorphism can be lifted uniquely to an isomorphism of simplicial abelian groups $\varphi : A_{\bullet } \simeq A_{\bullet }^{\operatorname{op}}$. The isomorphism $\varphi $ is characterized by the requirement that for every $n$-simplex $x \in A_ n$, we have $\varphi (x) \equiv (-1)^{n} x$ modulo degenerate simplices of $A_{\bullet }$.