Contravariant functor

A contravariant functor (as opposed to a covariant functor) behaves very much like a covariant functor, except for the fact that it flips morphisms and reverses composition. Because of this, a contravariant functor $𝐹$ from $𝖢$ to $𝖣$ is equivalent to a covariant functor from the Opposite category of $𝖢$ .

Notation

In these notes, the notation for functors will always be covariant. We will refer to a functor
$𝐹 : 𝖢 𝐨 𝐩 \to 𝖣$
as contravariant, even though we really mean we have a contravariant functor from $𝖢$ to $𝖣$ .

A special kind of contravariant functor is the Presheaf.

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Contravariant functor

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