Suppose we have four smooth maps between smoot manifolds:
$f: M \rightarrow X$ $g: X \rightarrow N$ $h: M \rightarrow Y$ $i: Y \rightarrow N$
an the equation on compositions of jets
$j_m(g \circ f) = j_m(i \circ h)$
Then are there allways representatives f' \in j_mf g' \in j_xg h' \in j_mh i' \in j_yi
with $f(m)=x$ and $h(m)=y$ and
(g \circ f)(m') = (i \circ h)(m')
for all m' on a neighbourood of $m$ ?
I guess it is yes but I can't see how to proof it.