In Japanese mathematics, the word
‘[functor](http://en.wikipedia.org/wiki/Functor)’ is translated as
[関手【かんしゅ】](http://ja.wikipedia.org/wiki/%E9%96%A2%E6%89%8B). What is the
etymology of this word?

I suspect that it is a pun on
[関数](http://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0_%28%E6%95%B0%E5%AD%A6%29)
(function). This leads to two further questions:

  1. Why 関手 and not 函手? According to Wikipedia 函数 was standard until the late 1950s. Category theory (圏論) was invented in the late 1940s, and was certainly known in Japan in the 1950s, c.f. Nobuo Yoneda's (米田信夫) work.
  2. Why -手? It _seems_ to be the same agentive morpheme -手 as in 運転手、歌手、選手 etc., but these are words referring to _people_. Moreover (to my knowledge) -手 is not as productive as, say, -者. Either way, it seems unnatural to calque _-or_ like this.

To ward off any confusion, I _do not_ mean 関手 as in せきて.
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etymology mathematics
The etymology of 関手【かんしゅ】
I do not have a definitive answer to either of your questions, but let me post
my thoughts anyway because definitive answers may be hard to obtain.

As for 1, kanji 函手 is also used, for example, in 圏論の基礎, the Japanese
translation of _Categories for the Working Mathematician_ written by Saunders
Mac Lane and translated by 三好博之 and 高木理 (translation published from Springer
in 2005). I do not know which of 関手 and 函手 is more common among
mathematicians, or how it compares to 関数 vs 函数.

As for 2, I can only speculate. I feel that -者 is associated with the notion
of physical people more closely than -手, and that using -者 to a non-human
object is a clear personification, which sounds unprofessional. But I am not
sure about this.
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1. Well, the 関 in 関数, 関手 is a replacement for 函, chosen to simplify the character set in usage. How logical would it be to simplify half of the vocabulary? Especially closely related vocab…

  2. I'm not sure it's a pun. 手 has the meaning of 取る, and a functor is actually a map from a category (a box, 函, since the Japanese word for function comes from a "box for numbers") to another. It is possible that 関手 thus means that it "takes a box('s contents)" to another box.

However, now that I think about it, "functor" is very close to "function" too…
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This is a great question. I searched the Iwanami mathematical dictionary
『岩波数学辞典』 and Sasahara's 当て字 dictionary, 『当て字・当て読み 漢字表現辞典』, and did not find a
definitive answer. Here's what I did find, though:

The word 函数 was invented in China, not Japan. The characters were chosen for
phonetic value as well as meaning. It's possible that "box that numbers go
into" was the intended interpretation, but we can't know for sure. What we can
know is that it was the closest the translators involved could get to the
"fun-" sound. [Edit: If I had bothered to read the Wikipedia article, I would
know that it said that 函 was chosen as the /han/ sound for its meaning of
"enclose, contain", and is apparently used in the sense that that the
independent variable (x) is included within the dependent one (y). The "black
box" interpretation is specifically rejected.]

The spelling 関数 was created in Japan once the postwar kanji and vocab lists
([tōyō kanji](http://en.wikipedia.org/wiki/T%C5%8Dy%C5%8D_kanji),
[学術用語集](http://ja.wikipedia.org/wiki/%E5%AD%A6%E8%A1%93%E7%94%A8%E8%AA%9E%E9%9B%86))
were created without 函 on them. 関数 is ateji upon ateji. The two forms would
have coexisted for a while before 関数 became dominant. As Tsuyoshi's answer
says, this means that we can basically ignore the question of 関 vs 函; official
kanji lists aren't able to cause sudden and discrete changes in everyone's
usage, so it's no big deal if we see some inconsistency and overlap.

What follows is pure speculation.

  * If "functor" was coined in the 1940s, then 関手 was probably coined in Japan, because by that period Western science was not absorbed via China.
  * The English word is the root of "function" plus the Latinate suffix -or which means "(male) person who performs a task." Actor, director, operator, etc.
  * So maybe whoever coined the word 関手 took the first half of 関数 and added the 手 of 運転手, 歌手, etc., by specific analogy with -or.

Basically I am in agreement with your theory and offering a reason why a
suffix normally associated with people would have been chosen: because that's
how the English breaks down, too. It may have been the very fact that 手 is no
longer productive that made it the choice for calquing -or: as Tsuyoshi says,
using a productive suffix like 者 might have carried too strong an implication
of "(actual) person."
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>   2. Why -手 ?
>

I can't think of -手 used in this way for other math, physics, ... terms.

In terms of consistency, 関子 or 函子 is (would have been) probably better.

演算子 論理演算子 比較演算子 モナド変換子 (monad transformer) 拡張子 変換指定子 変換子 変換修飾子 z変換演算子 ラプラス変換子
ラプラス変換演算子 フーリエ変換演算子 座標変換演算子 微分演算子

> <https://ja.wikipedia.org/wiki/%E5%AD%90>
>
> 西洋語の翻訳語の一部。
>
>   * 生物学の用語。遺伝子（gene）、精子（sperm）、配偶子（gamete）、胞子（spore）、卵子（ovum）など。
>
>   * ギリシア語由来の接尾辞 -on や動作主体を指す -or
> に相当する語。電子（electron）、重力子（graviton）、接続子（connector）、演算子（operator）、など。
>
>
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