Thank you very much for the answer [Jonathan Castello](—it was very helpful!
If I understood correctly, when looking for the adjoint of a function we can start from the formula and, if it's well defined, we still have to prove the definition of adjoints.
Otherwise, if the formula is not well defined or we are able to come up with a counter-example of the definition, then there is no adjoint.
I'm still not very confident on how to come up with counter-examples, but I guess this is a matter of experience.

I would have loved to see puzzles 18 and 19 solved in a more mechanical fashion, that is, starting from the formulas and applying the corresponding definitions.
In particular, for puzzle 19, I would like to see how we arrive at [the answer posted by David Tanzer](