I wrote:

> So, here's what we've shown:

> If \$$f : A \to B\$$ has a right adjoint \$$g : B \to A\$$ and \$$A\$$ is a poset, this right adjoint is unique and we have a formula for it:

$g(b) = \bigvee \\{a \in A : \; f(a) \le_B b \\} .$

> And we can copy our whole line of reasoning and show this:

> If \$$g : B \to A\$$ has a left adjoint \$$f : A \to B\$$ and \$$B\$$ is a poset, this left adjoint is unique and we have a formula for it:

$f(a) = \bigwedge \\{b \in B : \; a \le_A g(b) \\} .$

Jonathan wrote:

> My take is that the formula tells us what values the [...] adjoint must take if it exists.

That's true, and that's what I said in my lecture: I derived each of these formulas assuming the adjoint in question exists.

But I think the situation is much better than this. I believe that if the formulas give well-defined functions, these are necessarily the desired adjoints! It would be fun to try to show this, straight from the definitions. My hunch is that it's easy.