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We show that the following conditions are equivalent: (1) $\\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \\in \\mathcal{C}$ is the unit object. We also give some other equivalent conditions. 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We show that the following conditions are equivalent: (1) $\\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \\in \\mathcal{C}$ is the unit object. We also give some other equivalent conditions. 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