Proposition 3.8. Assuming that:

$F : C \to D$ , $G : D \to C$ , $α : 1_{C} \to G F$ and $β : F G \to 1_{D}$ be an equivalence of categories as defined in Definition 1.9

Then there exist isomorphisms

α^{'} : 1_{C} \to G F

and

β^{'} : F G \to 1_{D}

satisfying the triangular identities. In particular,

(F ⊣ G ⊣ F)