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Let $(M,\otimes)$ be a small symmetric monoidal category. Is it possible to choose in each isomorphy class $[A]$ a representative $A_0$ and for each $A\in M$ an isomorphism $\phi_A:A\to A_0$ in a way that for any $A,B\in M$ the equation $$ \phi_{A\otimes B}=\phi_{A\otimes B_0}\circ(1\otimes\phi_B) $$ holds?

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I would try disproving that claim by taking the category to be the strict 2-group with two elements, each element with an automorphism group $\mathbb Z/2$ and all that equipped with the non-trivial associator $\alpha\in H^3(\mathbb Z/2,\mathbb Z/2)$. –  André Henriques May 20 '13 at 20:14
    
Sorry... this is not symmetric monoidal. –  André Henriques May 20 '13 at 20:17
    
Have you tried the category of finitely generated projective $R$-modules? –  Martin Brandenburg Jun 17 '13 at 17:04

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