I know that the functor from the category of open subsets of a manifold $M$ to the Sets, taking an open set $U$ and associating to it the collection of $C^k$ maps to $\mathbb{R}$ is a sheaf.
My question is a simple generalization; If $M$ and $N$ are manifolds and $O(M)$ is the category of open subsets of $M$ with morphisms being inclusions, then is the functor: \begin{equation} U \mapsto C^k(U,N) \end{equation} a sheaf also?
If so what is a good reference?