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.