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

Question

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.

Answer 1

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 $\mathrm{Hom}\,(FX,Y)\simeq\mathrm{Hom}\,(X,GY)$.

Answer 2

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.