Dan wrote:

> However, I'm not sure how I can obtain this sort of correspondence for the general case:
given a pair of adjoint functors \\(F \dashv G\\), I understand that there exists a one-to-one correspondence between \\(\mathcal{D}(F(A), B)\\) and \\(\mathcal{C}(A, G(B))\\),
but I don't know how to get a hold on the bijection \\(\phi : \mathcal{D}(F(A), B) \to \mathcal{C}(A, G(B))\\).

A pair of adjoint functors is, by definition, a pair of functors _equipped with_ a bijection of this sort, which is also required to be natural:

> **Definition.** Given categories \\(\mathcal{A}\\) and \\(\mathcal{B}\\), an **adjunction** is pair of functors \\(F: \mathcal{A} \to \mathcal{B}\\) and \\(G: \mathcal{B} \to \mathcal{A}\\) together with a natural isomorphism

> \\[ \phi : \mathcal{B}(F(-),-) \to \mathcal{A}(-,G(-)). \\] We call \\(F\\) the **left adjoint** of \\(G\\), and \\(G\\) the **right adjoint** of \\(F\\).

So, it's not just that _there exists_ a one-to-one correspondence between \\(\mathcal{D}(F(A), B)\\) and \\(\mathcal{C}(A, G(B))\\): rather, we only call \\(F\\) and \\(G\\) adjoint functors only if they are _equipped with_ a one-to-one correspondence which is natural. So, nobody can justly claim to have a pair of adjoint functors until they can hand you this bijection. It is not up to you to find it; it is up to them to provide it!

I could be missing the point of your concern - but if not you, surely some others need to think hard about the difference between structure (like _being equipped with_) and property (like _there exists_).

One way I could be missing your point is this. Maybe you're more concerned about how you find adjunctions in the first place.

If we are trying to _find_ a pair of adjoint functors between two categories, it won't just fall from the sky: we have to work for it. Usually we start by guessing two functors \\(F: \mathcal{A} \to \mathcal{B}\\) and \\(G: \mathcal{B} \to \mathcal{A}\\) that 'feel like adjoints' - for example, \\(F\\) feels like it's freely building objects of \\(\mathcal{B}\\) from objects of \\(\mathcal{A}\\), while \\(G\\) feels it's forgetting structure on objects of \\(\mathcal{B}\\) to get objects of \\(\mathcal{A}\\). That's true in this lecture's puzzles.

But then we have to dream up a bijection

\[ \phi_{A,B} : \mathcal{D}(F(A), B) \to \mathcal{C}(A, G(B)) \]

for every object \\(A\\) of \\(\mathcal{A}\\) and every object \\(B\\) of \\(\mathcal{B}\\). This is the key step. Usually we do this by studying \\(\mathcal{D}(F(A), B)\\) and \\(\mathcal{C}(A, G(B)) \\) and realizing that they are 'two ways of thinking about the same thing'.

And then - the most technical part, which we haven't dug into yet - we have to prove that \\(\phi\\) is natural. Luckily, this last part is just a calculation, which almost always works if our construction of it is 'systematic', not ad-hoc or requiring case-by-case choices.