I'm not sure this problem makes any sense on several levels, but here is the question verbatim:
Find natural coordinates for the double tangent bundle $TTM$. Show that there is a nice map $s:TTM \to TTM$ such that $s\circ s = \text{id}_{TTM}$ and such that $T\pi \circ s = T\pi_{TM}$ and $T\pi_{TM} \circ s = T\pi$. Here $\pi:TM \to M$ and $\pi_{TM}:TTM \to TM$ are the appropriate tangent bundle projection maps.
The first part is not an issue. The rest is what bothers me.
$T\pi \circ s = T\pi_{TM}$ does not make sense since the map on the left is from $TTM \to TM$ and the one on the right is from $TTTM \to TTM$.
$T\pi_{TM} \circ s = T\pi$ does not make sense either since the maps on the left do not even compose, and even if they did, the one on the left would be from $TTM \to TTM$ whereas the one on the right is from $TTM \to TM$.
I have been trying to see if there is any way to rectify the situation, but any adjustment in domains and codomains solves one problem but leads to another. Since I doubt this nonsense question could have made it into the textbook unless there was a typo, I was wondering if anyone sees how to make sense of this problem by fixing something.