To emphasize that one means a functor as stated and not as a functor one sometimes says covariant functor when referring to non-contravariant functors, for emphasis.
Equivalently, a contravariant functor from to may be thought of as a functor from to , but the version above generalises better to functors of many variables.
Also notice that while the objects of the functor category are in canonical bijection with those in the functor category (both are contravariant functors from to ), the morphisms in the two functor categories are in general different, as
This matters when discussing a natural transformation from one contravariant functor to another.
Since n-categories (and also (infinity,n)-categories) have different kinds of opposite category depending on which of the -morphisms are reversed for (see for instance opposite 2-category), they also have different kinds of “contravariant functor”.
Categories, covariant functors, and natural transformations form a 2-category Cat. To include the contravariant functors as well, we can equip with a duality involution, or we can generalize to a 2-category with contravariance, or some more general structure that also includes extranatural transformations or dinatural transformations. There could also be higher-categorical versions, such as a 3-category with contravariance.