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

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

This is made more precise below. The order "tensor-hom adjunction" is because tensor is the left adjoint, while hom is the right adjoint.

## General Statement

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

$\mathcal{C} = \mathrm{Mod}_R \quad \text{and} \quad \mathcal{D} = \mathrm{Mod}_S.$

Fix an (R,S) bimodule X and define functors F: CD and G: DC as follows:

$F(Y) = Y \otimes_R X \quad \text{for } Y \in \mathcal{C}$
$G(Z) = \operatorname{Hom}_S (X, Z) \quad \text{for } Z \in \mathcal{D}$

Then F is left adjoint to G. This means there is a natural isomorphism

$\operatorname{Hom}_S (Y \otimes_R X, Z) \cong \operatorname{Hom}_R (Y , \operatorname{Hom}_S (X, Z)).$

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.[1]

## 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

$\varepsilon : FG \to 1_{\mathcal{C}}$

has components

$\varepsilon_Z : \operatorname{Hom}_S (X, Z) \otimes_R X \to Z$

given by evaluation: For

$\phi \in \operatorname{Hom}_R (X, Z) \quad \text{and} \quad x \in X,$
$\varepsilon(\phi \otimes x) = \phi(x).$

The components of the unit

$\eta : 1_{\mathcal{D}} \to GF$
$\eta_Y : Y \to \operatorname{Hom}_S (X, Y \otimes_R X)$

are defined as follows: For y in Y,

$\eta_Y(y) \in \operatorname{Hom}_S (X, Y \otimes_R X)$

is a right S-module homomorphism given by

$\eta_Y(y)(t) = y \otimes t \quad \text{for } t \in X.$

The counit and unit equations can now be explicitly verified. For Y in C,

$\varepsilon_{FY}\circ F(\eta_Y) : Y \otimes_R X \to \operatorname{Hom}_S (X , Y) \otimes_R X \to Y \otimes_R X$

is given on simple tensors of YX by

$\varepsilon_{FY}\circ F(\eta_Y)(y \otimes x) = \eta_Y(y)(x) = y \otimes x.$

Likewise,

$G(\varepsilon_Z)\circ\eta_{GZ} : \operatorname{Hom}_S (X, Z) \to \operatorname{Hom}_S (X, \operatorname{Hom}_S (X , Z) \otimes_R X) \to \operatorname{Hom}_S (X, Z).$

For φ in HomS(X, Z),

$G(\varepsilon_Z)\circ\eta_{GZ}(\phi)$

is a right S-module homomorphism defined by

$G(\varepsilon_Z)\circ\eta_{GZ}(\phi)(x) = \varepsilon_{Z}(\phi \otimes x) = \phi(x)$

and therefore

$G(\varepsilon_Z)\circ\eta_{GZ}(\phi) = \phi.$

## References

1. ^ May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.