# Why are left/right adjoint functors not called up/down?

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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Why is anything in math not called anything else...? – Zev Chonoles Feb 9 '14 at 20:04
We do call it upper/lower adjoint... but only in order theory. – Zhen Lin Feb 9 '14 at 20:07
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... – Malice Vidrine Feb 9 '14 at 20:57
@JoeyBF: In your definition of an adjunction the naturality condition is missing. – Martin Brandenburg Feb 9 '14 at 23:31
@MartinBrandenburg Yes, I omitted it for brevity. My point was only about the placement of $F$ and $G$ in the equation. Nice catch though ;) – 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 $\mathrm{Hom}\,(FX,Y)\simeq\mathrm{Hom}\,(X,GY)$.