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

${\displaystyle {\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.}$ 

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:

${\displaystyle F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}}$ 

${\displaystyle G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}}$ 

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

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

This is actually an isomorphism of abelian groups. More precisely, if ${\displaystyle Y}$  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.^[1]

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

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

has components

${\displaystyle \varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z}$ 

given by evaluation: For

${\displaystyle \phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,}$ 

${\displaystyle \varepsilon (\phi \otimes x)=\phi (x).}$ 

The components of the unit

${\displaystyle \eta :1_{\mathcal {D}}\to GF}$ 

${\displaystyle \eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$ 

are defined as follows: For ${\displaystyle y}$  in ${\displaystyle Y}$ ,

${\displaystyle \eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$ 

is a right ${\displaystyle S}$ -module homomorphism given by

${\displaystyle \eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.}$ 

The counit and unit equations can now be explicitly verified. For ${\displaystyle Y}$  in ${\displaystyle {\mathcal {D}}}$ ,

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y}):Y\otimes _{R}X\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)\otimes _{R}X\to Y\otimes _{R}X}$ 

is given on simple tensors of ${\displaystyle Y\otimes X}$  by

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.}$ 

Likewise,

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}:\operatorname {Hom} _{S}(X,Z)\to \operatorname {Hom} _{S}(X,\operatorname {Hom} _{S}(X,Z)\otimes _{R}X)\to \operatorname {Hom} _{S}(X,Z).}$ 

For ${\displaystyle \phi }$  in ${\displaystyle \operatorname {Hom} _{S}(X,Z)}$ ,

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )}$ 

is a right ${\displaystyle S}$ -module homomorphism defined by

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)}$ 

and therefore

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .}$ 

The Hom functor ${\displaystyle \hom(X,-)}$  commutes with arbitrary limits, while the tensor product ${\displaystyle -\otimes X}$  functor commutes with arbitrary colimits that exist in their domain category. However, in general, ${\displaystyle \hom(X,-)}$  fails to commute with colimits, and ${\displaystyle -\otimes X}$  fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.

• Currying
• Ext functor
• Tor functor
• Change of rings

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9