Leinster gives a proof of the triangle identities (p. 52):
However, I'm trying not to use (2.3) directly (because there's no way to memorize those identities), instead, I'm trying to draw naturality squares (from which (2.3) follows). Here's what I have so far:
The problem is to show that the left vertical arrow $1_{F(A)}$ is equal to the composition of the vertical arrows $\epsilon_{F(A)}\circ F(\eta_A)$. There must be an easy way to show that this holds (I believe), but I don't see it. What am I missing?
The naturality squares you've drawn work for any pair of natural transformations with appropriate sources and targets, so it doesn't look like you're using the fact that ε and η come from an adjunction. In particular, you're not using the correspondence between hom-sets or the naturality of that correspondence.
That suggests that you won't be able to prove the triangle identities just from the diagram you've drawn at the end.
Is equation 2.3 ($\overline{f \circ p} = \bar{f} \circ Fp$) really that unmemorable? You could, for instance, remember the way it arises from the naturality of the correspondence by saying, "Precomposing followed by Transpose is the same as Transpose followed by Precomposing," and then just work out where you need to apply a functor to make the sources and targets match up in the latter part.
Or you could look ahead a bit, to where we reconstruct the correspondence from the unit and co-unit. If you can remember that the transpose of $f: A \rightarrow GB$ is given by $FA \xrightarrow{Ff} FGB \xrightarrow{\varepsilon_B} B$, you'll be able to reconstruct equation 2.3 from that.