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Remark Let $A_{\bullet }$ be a simplicial abelian group. Then Warning supplies a canonical isomorphism of normalized Moore complexes $\mathrm{N}_{\ast }(A) \simeq \mathrm{N}_{\ast }( A^{\operatorname{op}} )$. By virtue of Theorem, 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 $A_{\bullet }$.