Why are contravariant functors called contravariant? I'm just now learning a bit of category theory, and there often seems to be a certain notion, like limits for instance, and if you inverse certain arrows, you obtain a co-object related to that notion (for instance colimits).
Why is it, then, that we speak of contravariant functors instead of co-functors?
 A: The dual functor of a functor $F: \mathbb{A} \rightarrow \mathbb{B}$ is the functor $F^\mathrm{op}: \mathbb{A}^\mathrm{op}\rightarrow \mathbb{B}^\mathrm{op}$ defined in the obvious way. So, for a duality principle, we would not define co-functors as contravariant functors.
A: The definition of a functor is self-dual. If you reverse all the arrows in the definition of a functor $\mathsf{C} \to \mathsf{D}$, what you get is a functor $\mathsf{C}^\mathrm{op} \to \mathsf{D}^\mathrm{op}$, which is exactly the same thing as a functor $\mathsf{C} \to \mathsf{D}$. So in this sense a cofunctor is just a functor.
Now a contravariant functor is a functor too, but between different categories: a contravariant functor $\mathsf{C} \to \mathsf{D}$ is the same thing as a functor $\mathsf{C}^\mathrm{op} \to \mathsf{D}$, which is exactly the same thing as a functor $\mathsf{C} \to \mathsf{D}^\mathrm{op}$. This is rather unrelated to functors $\mathsf{C} \to \mathsf{D}$.
Note that some people do use the words "cofunctor" for contravariant functors, I guess it's just a matter of taste as long as every word is defined.
