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Other properties

If A is an abelian category and A is an object of A, then Hom (A, –) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if A is projective.

Let R be a ring and M a left R - module. The functor Hom (M, –): Mod - R → Ab is adjoint to the tensor product functor – ⊗ {\displaystyle \otimes } M: Ab → Mod - R.