Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product ${\displaystyle -\otimes X}$ and hom-functor ${\displaystyle \operatorname {Hom} (X,-)}$ form an adjoint pair:

${\displaystyle \operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).}$

This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.