What's the process by which one can find the left adjoint for a given monotone function?
For example, excercise 1.95 in Dr. Spivak and Dr. Fong's book "An Invitation to Applied Category Theory" asks if there exists a left adjoint mapping from the integers to the reals , for the monotone function ceil(x/3).
Apart from trying out a bunch of stuff, I'm having a hard time figuring out a method for systemically finding such a function. Playing around with the inequalities in the definition for left and right adjoints doesn't seem to net anything definitive, and even the proof in the book uses contradiction to prove one doesn't exist by trying out a couple of variables.
I'm assuming there is some proof that makes this easier perhaps, but I'm looking for the intuitive approach. Any ideas? Thanks