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People are used to the prefix co- flipping arrows in a concept1, and I have seen people using cofunctor to mean a functor that flips arrows, i.e. that takes $A \to B$ to $FB \to FA$. I know this concept by the name contravariant functor, which I believe is standard, and contrasted with a covariant functor which doesn't flip arrows – the latter making the name cofunctor particularly unhelpful.

Nevertheless, Wiktionary and Wikipedia (last sentence in the contravariance paragraph) both back up this usage, though uncitedly, so it seems to have some traction. Is there any substance to this usage? Are there any authors who support it, or mention it?

If it is just incorrect, has anyone authoritative dismissed it in a citeable way, such that it could be banished from Wikipedia in future?

Edit: I've also heard contrafunctor used to mean a contravariant functor. Does this mean there are people who use cofunctor to mean covariant functor?

1 What do I mean by "concept" here? Well, an ncept with the arrows flipped, of course.

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Cofunctor is used in this sense in the older algebraic topology literature, e.g. in Husemöller's Fibre bundles. – Martin May 17 '13 at 13:34
I think "covariant" is the outlier here, and is using the co- prefix differently. "Contrafunctor" appears to be a contraction of "contravariant functor" and isn't used by mathematicians that I'm aware of. At any rate, there isn't necessarily a tidy duality here, because to make sense of arrows being "flipped" requires a context establishing a preferred direction for arrows, which a functor in full generality doesn't have. – camccann May 17 '13 at 13:55

I might be wrong, but I don't think it is completely right to call a controvariant functor "cofunctor" (at least if we want to stick to the convention of using the particle co- to mean and evoke duality) because the sentence "$T$ is a functor" is clearly self-dual, as Mac-Lane explicitely pointed out in his Categories for the Working Mathematician. So, given a functor $T$ a "cofunctor", interpreted as the dual concept, would be just $T$ itself.

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Suppose we're working in a sufficiently big universe such that both A and B belong to that universe. Then "$T$ is a functor from $A$ to $B$" may be dualize to obtain "$T$ is a functor from B to A", if we think of it as an arrow of Cat. Don't you agree? – Edoardo Lanari May 17 '13 at 22:40
The only possibility is that you mean "T is a functor" is self-dual, without explicitly mentioning its domain and codomain. Then I agree with what you meant. – Edoardo Lanari May 17 '13 at 22:50
Yes, you're probably right: what I really meant was that axioms defining a functor are self-dual (reversing arrows definitely doesn't change them and logical quantifiers are not involved in the dualizing process), so also the statement "T is a functor" is. I probably was a little bit careless: I'll edit! – Marco Vergura May 17 '13 at 23:03
Even if, quoting Mac-Lane: "Duality for a statemen involving several categories and functors between them reverses the arrows in each category but does not reverse the functors" – Marco Vergura May 17 '13 at 23:10
@StevenShaw Sure, thank you for spotting it! – Marco Vergura Sep 6 '15 at 2:27

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