Suppose $F\colon\mathcal{A}\to\mathcal{B}$ and $G\colon\mathcal{B}\to\mathcal{A}$ are adjoint functors between some $R$-linear abelian categories ($R$ a ring), via a fixed counit-unit adjunction $\epsilon\colon GF\Rightarrow 1$ and $\eta\colon 1\Rightarrow FG$. Why is the following an isomorphism of $R$-algebras?
$$ \operatorname{Nat}(G,G)\to\operatorname{Nat}(F,F):\alpha\mapsto (F\ast\epsilon)\circ (F\ast\alpha\ast F)\circ (\eta\ast F) $$
My primary issue is I don't see how to invert it, since the counit-unit equations $F\epsilon\circ \eta F=1$ and $\epsilon G\circ G\eta=1$ don't seem to allow me to "undo" the $F\ast\epsilon$ and $\eta\ast F$ from the left and right.