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We consider the category $\\mathcal{Cob}_G$ of $3$-dimensional cobordisms equipped with a representation of their fundamental group in $G$, and the category ${Vect}_{\\mathbb{F},\\pm G}$ of $\\mathbb{F}$-linear maps defined up to multiplication by an element of $\\pm G$. Using the elementary theory of Reidemeister torsions, we construct a \"Reidemeister functor\" from $Cob_G$ to ${Vect}_{\\mathbb{F},\\pm G}$. 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