Not sure exactly what the two morphisms "are" in **Puzzle 168** but I just noticed this which may or may not help:

$$\alpha_{a',b'}(g \circ h \circ F(f)) : a' \rightarrow G(b')$$

and

$$G(g) \circ \alpha_{a,b}(h) \circ f : a' \rightarrow a \rightarrow G(b) \rightarrow G(b')$$

The naturality square essentially went around the world to say these two are equal.