# What does $\Rightarrow$ in the definition of a natural transformation mean?

In the definition of a natural transformation from http://ncatlab.org/nlab/show/natural+transformation

it is said that a natural transformation is $\alpha : F \Rightarrow G$ (and the definition continues)

But what does that $\Rightarrow$ mean, how is it called?

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I read that as "maps to"... – J. M. Aug 21 '11 at 11:57
Do you mean $\mapsto$ or $\to$? – Yrogirg Aug 21 '11 at 12:00
It's called a arrow/morphism in the functor category if you insist on giving it a name. @J.M. I read that as "$\alpha$ is a natural transformation from $F$ to $G$." or simply "$\alpha$ from $F$ to $G$" – t.b. Aug 21 '11 at 12:09
I think the use of $\Rightarrow$ for natural transformations is more common among higher categorists, who think of them as $2$-morphisms in the $2$-category $\textbf{Cat}$. – Zhen Lin Aug 21 '11 at 14:39
@Zhen: that's a very good point. In fact, I don't know where the notation comes from but it is pretty standard in homological algebra nowadays even without mention of $2$-categorical ideas. I checked the usual suspects and the first reference I've found up to now using the $\Rightarrow$ notation is Bénabou's Introduction to bicategories. It doesn't appear that it was used earlier by any of the founding fathers Eilenberg-Mac Lane, Ehresmann, Freyd, Gabriel, Grothendieck, Kan, etc. – t.b. Aug 21 '11 at 22:27

## 1 Answer

Just look in the comments, especially the one by t.b., $\Rightarrow$ is just the notion of natural transformation, nothing more general.

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