If one defines a smooth $(r,s)$ tensor field on a differentiable manifold as a $C^{\infty}(M)$ multilinear map $$T : (\Gamma(T^*M))^r \times (\Gamma(TM))^s \xrightarrow{\sim} C^{\infty}(M)$$ from the modules of smooth sections of the cotangent and tangent bundles respectively, how does one define contractions (in a coordinate free way)? Must an approach with tensor products be employed?
I know the idea of contraction is to take a vector field input and covector field input and let the second act on the first to multiply the tensor by a smooth function, but I'm not sure how to implement it in this picture.
Approach via tensor products: Coordinate-free notation for tensor contraction?
The answer to that post gives a coordinate free way to get a contraction over a single vector space in terms of the universal property of the tensor product, similar to the idea I quoted above. So this is only applicable if we consider tensors as sections of the tensor bundle and hence are able to write in some coordinate patch on $M$ in a form like $$T = T_{\mu\nu} dx^\mu \otimes dx^\nu$$
So my question becomes: can one define tensor field contraction without going to a pointwise construction?
