Given two modules and over a unit ring , denotes the set of all module homomorphisms from to . It is an -module with respect to the addition of maps,
(1)
|
and the product defined by
(2)
|
for all .
denotes the covariant functor from the category of -modules to itself which maps every module to , and maps every module homomorphism
(3)
|
to the module homomorphism
(4)
|
such that, for every ,
(5)
|
A similar definition is given for the contravariant functor , which maps to and maps to
(6)
|
where, for every ,
(7)
|