I'm still unsure about a few things,
so I would really appreciate some feedback on my line of reasoning:

We can use the forumla to compute the right (or left) adjoint _only if_ we know that the right (or left) adjoint exists.
If the posets in question have all meets and joins, that doesn't say anything about the existence of the right (or left) adjoint;
this was the case of [puzzle 18](https://forum.azimuthproject.org/discussion/comment/16490/#Comment_16490)—even though the powerset ordered by inclusion forms a complete lattice, [\\(f_!\\) doesn't always have a left adjoint](https://forum.azimuthproject.org/discussion/comment/16501/#Comment_16501).
It seems to me that in general proving the existence of the adjoint is rather difficult, so probably the formulas cannot be readily applied.