@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.