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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