Isomorphism through adjunction An adjunction $F \dashv G$ gives a morphism $\phi(f) : A \to G B$ to each morphism $f : F A \to B$. Does $\phi(f)$ have any special property if I know that $f : F A \to B$ is an isomorphism?

The only special properties it will have are those that the unit has, because the transpose of $\mathrm{id} : F A \to F A$ is precisely the unit $\eta_A : A \to G F A$. So, for example, the triangle identities imply that $\eta_{G A} : G A \to G F G A$ and $F \eta_A : F A \to F G F A$ are split monomorphisms.