I am trying to show that if $(F,G,\eta,\epsilon)$ is the data of an equivalence of categories $\mathcal{C}$ and $\mathcal{D}$, then $Hom_\mathcal{C}(x,y)$ and $Hom_\mathcal{C} (GF(x),GF(y))$ are isomorphic.
I have managed to show that it is injective, but I am struggling to show it is surjective. Could anyone provide a hint in the right direction?