TITLE: Is Lang's definition of a tensor bundle nonstandard?
QUESTION [6 upvotes]: Background
Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$).  Suppose we have a functor $\lambda:\mathfrak{A}^{op}\times\mathfrak{B}\to \mathfrak{C}$.  
Let $f:E'\to E$ be a morphism belonging to $\mathfrak{A}$, and let $g:F\to F'$ be a morphism belonging to $\mathfrak{B}$.  (Note: These are morphisms of topological vector spaces).  
Then we have a map $$\matrix{Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))\\
(f,g)\mapsto\lambda(f,g)}$$
We say $\lambda$ is of class $C^p$ if for all manifolds $U$, and any two $C^p$ morphisms $U\to Hom(E',E)$ and $U\to Hom(F,F')$, the composition $$U\to Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))$$
is also of class $C^p$.  (Note: We can replace $\mathfrak{A}$ and $\mathfrak{B}$ with categories of tuples to generalize this to several variables.  In fact, this is what we do below.)
It is not hard to show that this induces a unique functor $$\lambda_X:VB(X, \mathfrak{A})^{op}\times VB(X,\mathfrak{B})\to VB(X,\mathfrak{C}).$$  on vector bundles taking values in the appropriate vector bundle categories over $X$.  
We define a tensor bundle of type $\mathbf{\lambda}$ on $X$ to be $\lambda_X(TX)=\lambda_X((TX,\dots,TX),(TX,\dots,TX))$, where $TX$ is the tangent bundle.  
However, this doesn't agree with the definition given on Wikipedia or anywhere else I've looked.  
Questions

Is this terminology nonstandard?
Is the notion itself nonstandard?
If the terminology is nonstandard, but the notion is standard, does it have a different name?  
Is this definition useful?
Does this include more vector bundles as tensor bundles than the standard definition?

REPLY [5 votes]: Is the notion non-standard? As Emerton says, the answer is no, except perhaps in minor details.
Is the terminology non-standard (and more permissive than the usual notion of tensor)?  Yes, I'd say it is. 
Because Lang allows arbitrary categories of Banach spaces, his notion is very general; by taking the categories small, one needn't have much functoriality at all. For instance, in the case where $\lambda$ has just one, covariant input, we can take $\mathfrak{B}$ to be the category with one object,  $\mathbb{R}^n$, and morphisms $GL(n)$. Then $\lambda(\mathbb{R}^n)$ is a representation $V$ of $GL(n)$. On an $n$-manifold $X$, the resulting bundle $\lambda_X(TX)$ is usually known as an associated bundle. Form the principal $GL(n)$-bundle $Fr_X$ of frames (consisting of pairs of $x\in X$ with an isomorphism $T_x X \to \mathbb{R}^n$). Then $\lambda_X(TX)= Fr_X \times_{GL(n)}V$. 
The notion of associated bundle is one that differential geometers often find more convenient than Lang's, because it allows one to say exactly which structure groups of interest (here, $GL(n,\mathbb{R})$) and exactly which representations. 
A restrictive definition of tensor bundles allows those associated bundles where $V$ is a tensor product of copies of $\mathbb{R}^n$ and its dual. A broader (and I suspect fairly standard) definition allows subquotient representations of such - in particular, symmetric tensors and differential forms. But Lang apparently allows other things, e.g. $r$-densities (take the representation of $GL(n)\to \mathbb{R}^\times$ given by $g\mapsto |\det g|^r$).
Is the greater generality (compared to the standard notion of tensor) useful? In examples (e.g. densities) yes; in the abstract, not really - the interesting geometry is attached to specific classes of structure groups and their representations.