Contravariant Functor

up:: Functor

A contravariant functor is a functor which flips morphisms in the target Category. It’s usually denoted as

$$F:C^{op}\to D$$

where $C^{op}$ is the Opposite Category of $C$.

Thus, formally, for each $x,y\in C$, we have that

$$f\in Ho{m}_C(x,y)⟺F(f)\in Ho{m}_D(F(y),F(x))$$

i.e. morphisms are flipped upon $F$‘s influence.

Examples

• When talking about Vector Spaces and their relation to their Vector Space Duals, it can be proven that The functor between vector spaces and their duals is contravariant.

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References

• What is a Functor? Definitions and Examples, Part 2