What's the point of naturality in the definition of an adjunction?

Question

Two functors $F: →$ and $G: →$ are adjoint iff there is a family of isomorphisms \begin{align} η_{X, A}: (FX, A)\tilde{\to} (X, GA) \end{align} which is natural in $X$ and $A$ (i.e., $(F\_, \_)≅(\_, G\_)$ as bifunctors).

In my mind, this definition is “clearly useful”, because I've known it for years and I thus see it everywhere. However, when explaining this to someone with a strong "it's only useful if I can prove something with it"-way of looking at things, I could not find a reasonable explanation as to why we care about naturality in practice.

So: For which facts is it important that the family of isomorphisms is natural in $X$ and $A$? For instance, is there an obvious property of adjunctions that just wouldn't work without naturality?

Answer 1

Suppose you have an object $T$ of $\mathcal{C}$, objects $X,Y$ of $\mathcal{D}$ and morphisms $a\colon FT\rightarrow X$, $b\colon FT\rightarrow Y$ and $f\colon X\rightarrow Y$ in $\mathcal{D}$ such that $f\circ a=b$. Then, the bijection given by the adjunction translates $a$ and $b$ into morphisms $\eta(a)\colon T\rightarrow RX$ and $\eta(b)\colon T\rightarrow RY$ in $\mathcal{C}$, whereas applying $R$ also yields a morphism $Rf\colon RX\rightarrow RY$ in $\mathcal{C}$. Now, naturality is precisely the condition that $Rf\circ\eta(a)=\eta(b)$ in this scenario (and the same in the analogous scenario involving pre-composition instead). This then generalizes from commutative triangles (which is what the above example is) to more general diagrams. Intuitively, the adjunction should not only translate morphisms into morphisms, but it should translate commutative diagrams into commutative diagrams.

Now, as @MarkKamsma also mentions in the comments, the single most important property of adjoint functors is the left resp. right adjoints preserve colimits resp. limits. You can prove this by explicitly verifying the universal property and doing so precisely requires you to translate a commutative diagram (a cocone resp. cone) into another one through the adjunction. This is naturality "in action".

Answer 2

It's category theory; of course naturality is going to be useful. Unfortunately it is commonplace to simply never mention when things are natural, in theorem statements and proofs, or to never remark that naturality has been used, or to assume without checking (and sometimes the check is not easy) that everything is natural. It's therefore not too surprising you want to ask this question.

Without naturality I just have an abstract isomorphism; I don't get any insight into how $F,G$ reflect structures and properties of one category to the other. It would not be easy to go through a textbook on category theory and identify everywhere that naturality of adjunctions is used; once noted it is never again emphasised. "$\cong$" should usually be interpreted as a natural isomorphism. I conjecture you use naturality of adjunctions far more often than you think (in both variables!).

By the way, suppose $\Delta$ is natural in one variable but not the other. Then we can redefine one of the functors to be the same on objects but act differently on morphisms so that we get a truly natural adjunction. But we still need some naturality to start with, to get off the ground. There is an extension of this principle to parametric adjunctions.

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The (co)continuity of adjoints, as mentioned, requires naturality.

The naturality of the unit and counit and indeed the very reason we can make any computations with the unit and counit at all relies on naturality; naturality of the isomorphism which you call $\eta$ but I'd rather call $\Delta$ is equivalent to the assertion that, for an arrow $f:X\to GA$ we have $\hat{f}=\epsilon_AF(f)$ and for some $g:FX\to A$ we have $\hat{g}=G(g)\eta_X$. These formulas are extremely useful and oft-used, as are their consequences such as the triangle identities, but require naturality of $\Delta$. The terminal/initial morphism perspective is about naturality of $\Delta$ in one of the variables, in disguise. These are maybe the most important points.

The property that adjunctions can transfer natural transformations between functors in one category to natural transformations in the other requires $\Delta$ to be natural. The property that adjunctions induce an equivalence on subcategories requires $\Delta$ to be natural.

I have no more immediate examples off the top of my head, but although I haven't said very much the examples given are already of massive importance.