Suppose $F : C \to D$ is an equivalence of categories with quasi-inverse $G: D \to C$. Does the pair $(F,G)$ necessarily form an adjunction? How would I prove this?
I think I can use the unit-counit definition of adjunction. ( Let $\eta$ be the unit $id_C \to R\circ L$ and and $\epsilon$ be the counit: $\epsilon: L \circ R \to id_D$ )
and the fact that $L$ is a fully faithful functor if and only if $\eta$ is an isomorphism.
but I have no idea how to do this. Any help is appreaciated