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?
Best Answer
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.