This is made more precise below. The order "tensor-hom adjunction" is because tensor is the left adjoint, while hom is the right adjoint.
Fix an (R,S) bimodule X and define functors F: D → C and G: C → D as follows:
This is actually an isomorphism of abelian groups. More precisely, if Y is an (A, R) bimodule and Z is a (B, S) bimodule, then this is an isomorphism of (B, A) bimodules. This is one of the motivating examples of the structure in a closed bicategory.
Counit and Unit
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
given by evaluation: For
The components of the unit
are defined as follows: For y in Y,
is a right S-module homomorphism given by
The counit and unit equations can now be explicitly verified. For Y in C,
is given on simple tensors of Y⊗X by
For φ in HomS(X, Z),
is a right S-module homomorphism defined by