Let R be a commutative ring. A tensor category (ℭ, ⊗, I, a, r, ℓ) is said to be a tensor R-category if ℭ is an R-category and if the tensor product functor is an R-bilinear map Hom(A, B)×Hom(C, D)->Hom(A⊗C, B⊗D) on sets of morphisms defined by (f, g)↦f×g.

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