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We make this relation precise by constructing an explicit tensor isomorphism between the representation category of $C(A)$ and the representation category of the affine symplectic group $\\operatorname{ASp}(A)=\\operatorname{Sp}(V_A)\\ltimes\\widehat{V_A}$. Thus $C(A)$ and $\\operatorname{ASp}(A)$ are isocategorical, although they need not be isomorphic. 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