I'm just curious about constructions like differential forms, as far as I can see, one always starts with a manifold $M$, some functor-like $F$ from vector spaces to vector spaces, e.g. $V\mapsto \bigwedge^2 V\otimes V^*$, and then constructs a new manifold where a coordinate patch $U$ becomes $U\times F(\mathbf{R}^n)$ and a coordinate transform $\phi:U\cap V\to U\cap V$ becomes $\phi\times F(D_\phi)$, if I understand correctly.
So I wonder, what does $F$ need to satisfy in order to make this work?
Also, what's the exact nature of something like $\bigwedge^2 V\otimes V^*$, it's not a functor, although, given $f:V\to W, g:W\to V$, I think one can construct a linear map from $\bigwedge^2 V\otimes V^*$ to $\bigwedge^2 W\otimes W^*$.
$F$ needs to be a continuous functor in the sense that if $V$ is a finite-dimensional real or complex vector space then the induced map $GL(V) \to GL(F(V))$ is continuous with respect to the Euclidean topology. This means that $F$ sends cocycles to cocycles continuously which is all you need for $F$ to send vector bundles to vector bundles. In typical examples like exterior and symmetric powers $F$ is even polynomial, and the same will be true more generally for any Schur functor. An example of a continuous $F$ which is not polynomial is $|\det(V)|$ which is used to construct the density bundle.
$\Lambda^2(V) \otimes V^{\ast}$ is a functor with respect to isomorphisms and that's all we need here. We can also consider functors of two or more variables and there the condition is that the induced map $GL(V) \times GL(W) \to GL(F(V, W))$ is continuous etc., which gets the direct sum and tensor product into the game as well.