Other Properties
If A is an abelian category and A is an object of A, then HomA(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 HomZ(M,–): Ab → Mod-R is right adjoint to the tensor product functor – R M: Mod-R → Ab.
Read more about this topic: Hom Functor
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