I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this:
We have an operator that acts on vectors in the tangent space and, in order to show that it "lives at points", we generalise it to operate on global sections in vector bundles. Then, by showing this operator is linear over $C^\infty$ functions, some argument involving bump functions implies the operator defines a tensor field.
Usually this is followed by some remark about modules, giving a strong impression that something bigger is going on.
On general relativity texts, there's an equivalent practice that involves showing the operator "transforms correctly".
This is all quite vague, since I cannot quite point my finger at what's the general phenomenon. So my question is:
What is the relation between tensor fields and $C^\infty$ linearity and what does this has to do with modules?