Functor
From AoPSWiki
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 .
Which satisfies the conditions:
-
for all 
.
-
for all morphisms and 
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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