nLab
contravariant functor

A contravariant functor $F$ from a category $C$ to a category $D$ is simply a functor from the opposite category $C^{\mathrm{op}}$ to $D$.

To emphasize that one means a functor $C\to D$ as stated and not as a functor $C^{\mathrm{op}}\to D$ one sometimes says covariant functor for non-contravariant, for emphasis.

Equivalently, a contravariant functor from $C$ to $D$ may be thought of as a functor from $C$ to $D^{\mathrm{op}}$, but the version above generalises better to functors of many variables.

Also notice that while the objects of the functor category $[C^{\mathrm{op}},D]$ are in canonical bijection with those in the functor category $[C,D^{\mathrm{op}}]$ (both are contravariant functors from $C$ to $D$), the morphisms in the two functor categories are in general different, as

$$[C^{\mathrm{op}},D]\simeq [C,D^{\mathrm{op}}{]}^{\mathrm{op}}.$$[C^{op}, D] \simeq [C, D^{op}]^{op} \,.

This matters when discussing a natural transformation from one contravariant functor to another.