Suppose $\left \langle F,G,\eta ,\varepsilon \right \rangle$ is an adjunction. It is easy to show that if $G$ is full, then $G\varepsilon $ is invertible with inverse $\eta G$. But MacLane says that this is also true if $F$ is full. I do not see how this works though. I have an $\alpha :GFGa\rightarrow Ga$ s.t. $F\alpha = \varepsilon _{FGa}$ and then $\alpha \cdot \eta _{Ga} = id_{Ga}$ but do not see how to proceed from here.
edit: Following the kind hint given in the comments, the result follows by:
1). If $G$ is full, $G\varepsilon $ is invertible
2). If $F$ is full, $F \eta $ is invertible (the proof of which is basically the math in the last sentence of the first paragraph).
3). $G\varepsilon $ is invertible $\Leftrightarrow F \eta $ is. This is an easy application of the triangle identities.