Functor
A functor is a type of map between two categories.
More precisely, a functor 
is a mapping which
- sends every object
of 
to and object 
of 
. - sends every morphism
of to a morphism 
of .
of 
.
of 
of Which satisfies the conditions:
for all 
.
for all morphisms and 
of .
for all 
.
of A contravariant functor a mapping satisfying the same properties as above, except that 
is a morphism from 
to 
, and instead of having 
we have 
. Alternatively, we can define a contravariant functor as an ordinary functor 
, where 
is the opposite category of 
. We sometimes call our original type of functors covariant functor to distinguish them from contravariant functors.
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