I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and $\mathcal{D}$ and functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ then $F$ is left adjoint to $G$ iff

$$ \forall X \in C, \forall Y \in D, \hom_D(FX,Y) \cong \hom_C(X,GY) $$

And we see that $F$ appears in the left of the left hand side. I also learned the saying that left adjoints round up and right adjoints round down, in the sense that they add/forget additional structure. It seems to me that this viewpoint is much more practical to a working category-theorist than the rather technical Hom-set definition. My question is then, why are left/right adjoints not called up/down or top/bottom adjoints? It would seem much more natural, to me anyway.

As an example and a side question, how do you remember that forgetful functors are right adjoint and free ones left adjoint? I always get mixed up between the two. This is a nice example of why I think "forgetful functors are down-adjoint and free ones up-adjoint" would be more useful, to the beginner at least.

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    $\begingroup$ Why is anything in math not called anything else...? $\endgroup$ – Zev Chonoles Feb 9 '14 at 20:04
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    $\begingroup$ We do call it upper/lower adjoint... but only in order theory. $\endgroup$ – Zhen Lin Feb 9 '14 at 20:07
  • $\begingroup$ Personally, I learned about adjoints from an archaic source that used the terms "adjoint" and "co-adjoint". I actually still prefer this; I just have to remember what "adjoint" means, and then the other one's the dual of that :p But thankfully I did get the hang of the left-right terminology eventually... $\endgroup$ – Malice Vidrine Feb 9 '14 at 20:57
  • $\begingroup$ @JoeyBF: In your definition of an adjunction the naturality condition is missing. $\endgroup$ – Martin Brandenburg Feb 9 '14 at 23:31
  • $\begingroup$ @MartinBrandenburg Yes, I omitted it for brevity. My point was only about the placement of $F$ and $G$ in the equation. Nice catch though ;) $\endgroup$ – JoeyBF Feb 9 '14 at 23:43

First, adjoint functors do not always add/forget structure. For example, equivalences of categories are adjoint pairs, but these certainly do not always add or forget structure in any obvious way. The reason for labeling them left/right adjoints is exactly the reason you mention: because the equation $\mathrm{Hom}\,(FX,Y)\simeq\mathrm{Hom}\,(X,GY)$ is incredibly useful. If we called them up/down functors, then I'd have to perpetually consult wikipedia to remember which one appeared on the left and which one appeared on the right in the equivalence $\mathrm{Hom}\,(FX,Y)\simeq\mathrm{Hom}\,(X,GY)$.

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    $\begingroup$ Well, an equivalence does forger information in the non-strict sense, just like constant functions are increeasing :-) $\endgroup$ – Mariano Suárez-Álvarez Feb 9 '14 at 20:06
  • $\begingroup$ Equivalences also are not always adjoint pairs. Perhaps it would have been better to remark that an isomorphism of categories and it's inverse always form an adjunction, indeed one in which each functor hardly does anything to reach object. $\endgroup$ – Ittay Weiss Aug 2 '17 at 16:41
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    $\begingroup$ @IttayWeiss I don't understand your comment; the two functors forming an adjunction are always left and right adjoint to one another. $\endgroup$ – Arnaud D. Oct 12 '17 at 9:32

1) Lefts adjoints round up and right adjoints round down is not a very precise statement. What would you say then of a functor that is both a left and right adjoint? it rounds up and down at the same time? What you call the technical definition of adjunction (I don't see it as particularly technical, and once you will get used to it only a bit more I hope you will agree) captures precisely what the adjunction relationship is all about. The convention for hom sets then naturally gives rise to the left/right adjectives.

2) Free constructions are by definition left adjoints to forgetful functors. If you are unsure which one is which, then just try to establish the bijection and see which one works. Pretty soon you'll be doing such things in your head.

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    $\begingroup$ "To forget is right." $\endgroup$ – Berci Feb 10 '14 at 1:53
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    $\begingroup$ but to free is also right. And what is left behind is forgotten. $\endgroup$ – Ittay Weiss Feb 10 '14 at 1:55
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    $\begingroup$ To free is a very liberal thought. Very left. $\endgroup$ – k.stm Aug 1 '17 at 18:56

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