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)
|
