I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.

- the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, for any $f\in C^{\infty}(M,N)$;
- the map $(\alpha,\beta):J^k(M,N)\to M\times N$, defined by $(j^k_x f)\mapsto (x,f(x))$, is a smooth summersion, for any smooth manifolds $M$ and $N$;
- the composition map $\gamma:J^k(N,O)\times_{N} J^k(M,N)\to J^k(M,O)$, defined by $(j^k_{f(x)} g,j^k_x f)\mapsto j^k_x(g\circ f)$, is smooth, for any smooth manifolds $M$,$N$ and $O$, (here $J^k(M,N)\times_{N} J^k(M,N)$ is the fiber product of $\beta:J^k(M,N)\to N$ and $\alpha:J^k(N,O)\to N$);
- the map $(\alpha,\beta)^{-1}(U,V)\to J^k(U,V)$ defined by $j^k_x f\mapsto j^k_x(f|_{U\cap f^{-1}(V)})$ is a smooth isomorphism, for any open subsets $U\subset M$, $V\subset N$;
- for any open subsets $U\subset \mathbb{R}^m$ and $V\subset \mathbb{R}^n$, the map $J^k (U,V)\to U\times V\times \bigoplus_{i=1}^k{L^i_{sym}(m,n)}$, given by $j^k_x f\mapsto (x,f(x),Df(x),\ldots,(D^kf)(x))$, is a smooth isomorphism, (here $L^i_{sym}(m,n)$ is the vector space of the $\mathbb{R}^n$-valued symmetric $k$-multinear maps on $\mathbb{R}^m$).

Probably it is not sufficient, or redundant, but, in such a case, I would know if there is in the literature such a kind of characterization.

My question is: Once prescribed the usual smooth structure on the $J^k(U,V)$, for arbitrary open sets in euclidean spaces $U$ and $V$ (as in point 5), what kind of conditions are sufficient to uniquely determine the usual smooth structure on $J^k(M,N)$ for all other smooth manifolds $M$ and $N$?