Contravariant Functor

A functor is said to be contravariant if its (co)domain is an opposite category. That is to say, suppose \(f: c \to c'\) in some category \(C\). A covariant functor \(F: C \to D\) satisfies \(Ff: Fc \to Fc'\), whereas a contravariant functor \(G: C^{\text{op}} \to D\) satisfies \(Gf: Gc' \to Gc\).

Pre-Sheaves

For some category \(C\), a contravariant set-valued functor \(F: C^{\text{op}} \to \text{Set}\) is said to be a pre-sheaf on \(C\).