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.