I was reading M,Golubitsky & V.Guillemin's book Stable Mappings and Their Singularities ,in theorem 2.7 (page 40),needs to prove the following fact:
If $h: Y \rightarrow Z$ is smooth, then $h_{*}: J^{k}(X, Y) \rightarrow J^{k}(X, Z)$ is smooth. If $g: X \rightarrow Y$ is a diffeomorphism, then $g^{*}: J^{k}(Y, Z) \rightarrow J^{k}(X, Z)$ is a diffeomorphism.
The book says it follows from the construction of a smooth structure on $J^k(X,Y)$.I try to prove it in detail taking the push forward for example,that is to show under the coordinate chart,The push forward is smooth.I don't know how to write it clear.