@IgnacioViglizzo : isn't it the other way round?

the right adjoint \$$g\$$ of \$$f\$$ of approximates the inverse of \$$f\$$ from above: \$$p\leq g(f(p))\$$, whereas a true inverse (if it existed) would bring \$$f^{-1}(f(p))\$$ down to \$$p\$$.

and

the left adjoint \$$f\$$ of \$$g\$$ approximates the inverse of \$$g\$$ from below\$$f(g(q))\leq q\$$, whereas a true inverse (if it existed) would bring \$$g^{-1}(g(q))\$$ up to \$$q\$$.

But this seems to go against John's characterization of right adjoints being conservative and left ones being "generous", so I may have made a mistake somewhere.