Is a contravariant functor a functor?

Question

By definition a functor associates a morphism $f:A \rightarrow B$ to a morphism $F(f):F(A) \rightarrow F(B).$ However, a contravariant functor associates $f$ to $F(f):F(B) \rightarrow F(A).$ So it seems that a contravariant functor is similar to a functor but can it be called a functor?

Answer 1

A contravariant functor $F:C\rightarrow D$ can be viewed as a functor from $C^0\rightarrow D$ where $C^0$ is the opposite category of $C$.

Answer 2

You can formulate a notion of "contravariant functor" that's defined similarly to a (covariant) functor except the composition flips, but you shouldn't. Instead, you should only use the concept of (covariant) functor. Every "contravariant functor" can be identified with a covariant functor from an opposite category (or, equivalently, into an opposite category). That is, a "contravariant functor" from $\mathcal{C}\to\mathcal{D}$ is the same as a (covariant) functor $\mathcal{C}^{op}\to\mathcal{D}$ which is essentially the same as a functor $\mathcal{C}\to\mathcal{D}^{op}$. Thus, the notion of a "contravariant functor" is unnecessary.

Not only is it unnecessary, in my opinion which I believe is widely shared, it just adds confusion. While the term "contravariant" is still widely used, it is usually used when the source category is not explicitly named. For example, "presheaves are contravariant $\mathbf{Set}$-valued functors". Few nowadays would say "consider a contravariant functor $F : \mathcal{C}\to\mathbf{Set}$" when they meant a functor $\mathcal{C}^{op}\to\mathbf{Set}$. Indeed, it's more likely that one would say "consider a contravariant functor $F : \mathcal{C}^{op}\to\mathbf{Set}$" for the above, though this should be avoided, just say "consider a functor ...". Nevertheless, this is one potential source of confusion. Another source of confusion is that you have to remember that a "contravariant functor" is contravariant (and thus not a functor) which is information that is easily lost. For example, if I write $F : \mathcal{C}\to\mathbf{Set}$ on a board and tell you it's a "contravariant functor", you have to keep the "contravariantness" in your head. Looking back at the board and seeing $F : \mathcal{C}\to\mathbf{Set}$ won't help. Closely related to this, you have to remember that you can't naively compose a "contravariant functor" with other functors or even other "contravariant functors". Or rather you can, but you need to keep track of the "variantness" to know what kind of functor you end up with. Again, this is invisible information in a diagram like $\mathcal{C}\stackrel{F}{\to}\mathcal{D}\stackrel{G}{\to}\mathcal{E}$. And as before, considering only (covariant) functors and expressing contravariance through opposite categories this information can be made explicit, avoiding errors, e.g. $\mathcal{C}^{op}\stackrel{F^{op}}{\to}\mathcal{D}^{op}\stackrel{G}{\to}\mathcal{E}$.