Suppose that $A$ and $B$ are equivalent categories. Then there exist functors $F: A \rightarrow B$ and $G: B \rightarrow A$ such that $F \circ G \cong 1$ and $G \circ F \cong 1$. I am trying to show that $G$ is a left and right adjoint of $F$ (Actually, the problem states that these functors have left and right adjoints, but I think that these are the correct adjoints).
I have been trying to work with the definition of adjunction that two functors are adjoint when $B(F(A), B) \cong A(A, G(B))$. I haven't been able to work out isomorphism is that would show that these two fuctors are adjoint. How should I proceed?
$(Fa,b) \simeq (GFa,Gb) \simeq (a,Gb)$ The first bijection is due to $G$ being a category equivalence and the second is via composing with the natural transformation $Id \to GF $.