Trying to summarize what I get here as a pattern: If \$$f\dashv g\$$, then:

the right adjoint \$$g\$$ approximates the inverse of \$$f\$$ from above: \$$p\leq g(f(p))\$$

and

the left adjoint \$$f\$$ approximates the inverse of \$$g\$$ from below \$$f(g(q))\leq q\$$.

[Edited: I switched "right" and "left" in my first attempt, as Valter points out below. ]