Contravariant functor

Remarks

A contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ is the same as a functor $\mathcal{C}^{\mathrm{op}} \to \mathcal{D}$ from the opposite category, via $f^{\mathrm{op}} \mapsto F(f)$. Example: the power set with preimages, $A \mapsto \mathcal{P}(A)$ and $f \mapsto f^{-1}$, is a contravariant functor from sets to sets. Here $f^{-1} \colon \mathcal{P}(B) \to \mathcal{P}(A)$, and $(g \circ f)^{-1}(S) = f^{-1}(g^{-1}(S))$ gives clause 2. Another example is the dual space functor $V \mapsto V^*$, $f \mapsto f^*$ on vector spaces over a field, where $(g \circ f)^* = f^* \circ g^*$. Non-example: the power set with direct images, $f \mapsto (S \mapsto f(S))$, is not contravariant, because $f(\cdot)$ goes from $\mathcal{P}(A)$ to $\mathcal{P}(B)$, so clause 2 of the data fails; it is a covariant functor.