In order to define function spaces on manifolds one usually follows the same recipe and this begs for a categorial formulation. I would like to know whether there is a reference that discusses this in some detail.
Here is an example of what I have in mind: Let $E$ be a function space on $\mathbb{R}^d$, say $E=L^2(\mathbb{R}^d)$ or some Sobolev space. Then for $U\subset \mathbb{R}^d$ define $E_c(U)$ to consist of restrictions to $U$ of elements in $E$ with support compactly contained in $U$. Then $E_c(U)$ admits a natural Frechet space structure and for the examples above any diffeo $U\cong V$ of subsets of $\mathbb{R}^d$ induces an isomorphism $E_c(U)\cong E_c(V)$ of Frechet spaces (coordinate invariance).
Now if $M$ is a manifold, then one can define $E_c(U)$ whenever $U\subset M$ is a chart domain and for general $U$ via a partition of unity argument.
The two properties that allowed us to make the construction above are: 1) Elements of $E$ can be restricted, we can e.g. assume that $E$ embeds into the space of distributions on $\mathbb{R}^d$ and use the natural restriction there. 2) The local spaces $E_c$ are coordinate invariant.
I assume that there the construction can be formalised in the following way: For any function space $E$ with the properties 1) and 2) there exists a functor $\mathcal E_c: \mathsf{Man}\rightarrow \mathsf{LCTVS}$ from the category of smooth manifolds to the category of locally convex topological spaces such that $\mathcal{E}_c(U) \cong E_c(U)$ in a natural way when $U$ is an open subset of $\mathbb{R}^d$. What I am interested in is whether this in turn gives rise to a functor $E \mapsto \mathcal{E}_c$ with nice properties, e.g preserving compact embeddings, exact sequences and so on.
One very concrete example that I am interested in occurs in the setting of pseudodifferential operators. On $\mathbb{R}^d$ they are easy to define and one proofs e.g. that the symbol map gives rise to a short exact sequence of Frechet spaces. The same result is true on closed manifolds and I believe that this should follow from abstract nonsense.