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

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.

Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):

Fix an ${\displaystyle (R,S)}$-bimodule ${\displaystyle X}$ and define functors ${\displaystyle F\colon {\mathcal {D}}\rightarrow {\mathcal {C}}}$ and ${\displaystyle G\colon {\mathcal {C}}\rightarrow {\mathcal {D}}}$ as follows:

Then ${\displaystyle F}$ is left adjoint to ${\displaystyle G}$. This means there is a natural isomorphism

This is actually an isomorphism of abelian groups. More precisely, if ${\displaystyle Y}$ is an ${\displaystyle (A,R)}$-bimodule and ${\displaystyle Z}$ is a ${\displaystyle (B,S)}$-bimodule, then this is an isomorphism of ${\displaystyle (B,A)}$-bimodules. This is one of the motivating examples of the structure in a closed bicategory.

Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit

has components