# nLab contravariant functor

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

To emphasize that one means a functor $C \to D$ as stated and not as a functor $C^{op} \to D$ 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 $D^op$, but the version above generalises better to functors of many variables.

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

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

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

Revised on June 24, 2010 18:39:58 by Toby Bartels (75.117.107.247)