Suppose we have an equivalence of two categories $C$ and $D$, i.e., we have two functors $F: C \to D$ and $G: D \to C$ and natural isomorphisms $\alpha: 1_C \Rightarrow GF$ and $\beta: FG \Rightarrow 1_D$.
Now in principle we have two ways, in opposite directions, to connect $GFG$ and $G$: we could go through $$ \alpha G: G \Rightarrow GFG$$
or $$ G\beta: GFG \Rightarrow G.$$
Is there any chance these two maps are inverse to each other?
In general it won't be the case, but you can always change $\alpha$ and $\beta$ to make it true. This is basically the fact that if you have an equivalence then you have an adjoint equivalence. See for instance this question.
Yes, if the pair of functions $(L,R)$ and natural transformation $(\alpha,\beta)$ form an adjoint pair, then it is the case (it's one of the triangular conditions for the unit/counit of adjunction).
It's proven in "Categories for the Working Mathematician" that if you have an equivalence of categories, this can be made into an equivalence of categories that is also an adjunction (Theorem 1, page 93, Chapter 4, Section 4).
