Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to work without this assumption.

As described e.g. in Kriegl's and Michor's book, $C^\infty(M,N)$ is an $\infty$-dimensional manifold. Now I would like to construct a vector bundle (of rank $\infty$) whose fibre at $\varphi \in C^\infty(M,N)$ is given by $\Gamma(\varphi^\ast E)$. This should be possible because the fibres $\Gamma(\varphi_1^\ast E)$, $\Gamma(\varphi_2^\ast E)$ at homotopic maps $\varphi_1$, $\varphi_2$ are isomorphic. I have more or less an idea how to do the construction but have not worked out all the details needed to prove smoothness (of transition functions). However, the construction looks quite natural and I would almost expect that it has already appeared somewhere in the literature. Does anybody know a reference ? (I did some research but could not find anything ...)

**1. Update**(06.May 13)

As discussed below in the answer by Peter Michor, the total space of the bundle I am looking for is $C^\infty(M,E)$ and the (smooth) projection is $p_\ast : C^\infty(M,E) \rightarrow C^\infty(M,N)$.

However, I do not see that it actually defines *vector bundle* over the base $C^\infty(M,N)$, probably I am overlooking a piece of the argument. As far as I understand the proof of theorem 42.17 in the "Convenient Setting", it is shown that there exists a diffeomorphism $\Phi: TC^\infty(N,M) \cong C^\infty(M,TN)$ (I just neglect compact supports here) and $C^\infty(M,TN)$ inherits the structure of a vector bundle. In my case, I don't see that $C^\infty(M,E)$ is diffeomorphic to $T(something)$ and I did not find a "general" proof of such a vector bundle property for $C^\infty(M,E)$ in the Convenient Setting. Please let me know if there is any.

One idea to construct local trivializations for $C^\infty(M,E)$ might be given by identifying spaces $\Gamma(\varphi_1^\ast E)$ and $\Gamma(\varphi_2^\ast E)$ (for $\varphi_1,\varphi_2$ homotopic) using parallel transport w.r.t. some connection on $E$. I think this should be possible but proving that this construction depends smoothly on the basepoints $\varphi \in C^\infty(M,N)$ and does not depend on the choice of the connection seems to be technically complicated and I would like to avoid it if possible.