Can we get a soldering "form" whose pull-backs encode arbitrary tensor components? Context/notation: If $M^n$ is a smooth manifold, and $\pi\colon {\rm Fr}(TM) \to M$ denotes its frame bundle (it is a principal ${\rm GL}_n(\Bbb R)$-bundle whose elements are pairs $(x,\mathfrak{v})$ with $x \in M$ and $\mathfrak{v}$ an ordered basis for $T_xM$), we have the soldering form $\theta \in \Omega^1({\rm Fr}(TM); \Bbb R^n)$, defined by $$\theta_{(x,\mathfrak{v})}(X) = [{{\rm d}\pi}_{(x,\mathfrak{v})}(X)]_{\mathfrak{v}},\qquad \mbox{for}
\quad X \in T_{(x,\mathfrak{v})}{\rm Fr}(TM),$$where $[\cdot]_{\mathfrak{v}} \colon T_xM \to \Bbb R^n$ takes a tangent vector to its column vector in $\Bbb R^n$ of components relative to $\mathfrak{v}$. If $U \subseteq M$ is open and $\mathfrak{e}$ is a local frame on $U$ (that is, a local section of ${\rm Fr}(TM)$), the pull-back $\mathfrak{e}^*\theta \in \Omega^1(U; \Bbb R^n)$ satisfies $$(\mathfrak{e}^*\theta)_x(v) = [v]_{\mathfrak{e}_x}$$for all $x \in U$ and $v \in T_xM$. It also has nice properties like $R_A^*\theta = A^{-1}\circ \theta$ for all $A \in {\rm GL}_n(\Bbb R)$, where $R_A\colon {\rm Fr}(TM) \to {\rm Fr}(TM)$ is the map "change of basis via $A$" (it's a consequence of the general formula $[v]_{\mathfrak{v}A} = A^{-1}[v]_{\mathfrak{v}}$).
This construction does not work replacing $TM \to M$ with an arbitrary vector bundle $E \to M$ because the derivative of the bundle projection takes values in tangent spaces to $M$, and not on the fibers of $E$.
Question. Now consider the tensor bundle $TM^{\otimes r} \otimes T^*M^{\otimes s}$. Since a basis for a vector space gives rise to a basis of any associated tensor space, doing this pointwise we get (with suggestive notation) a map $${\rm Fr}(TM) \ni (x,\mathfrak{v}) \mapsto (x, \mathfrak{v}^{\otimes r}\otimes (\mathfrak{v}^*)^{\otimes s}) \in {\rm Fr}(TM^{\otimes r}\otimes T^*M^{\otimes s}).$$
I was wondering if whether we get some object $\Theta$ (which apparently will not be a $1$-form) such that for each local frame $\mathfrak{e}$ on some open set $U \subseteq M$, $\mathfrak{e}^*\Theta$ (whatever this is) will satisfy $$(\mathfrak{e}^*\Theta)_x(T) = [T]_{\mathfrak{e}_x^{\otimes r}\otimes (\mathfrak{e}_x^*)^{\otimes s}}, \quad\mbox{for}\quad T \in T_xM^{\otimes r}\otimes T_x^*M^{\otimes s}$$
Understanding the case $(r,s) = (0,1)$ would already provide some insight. For the same reason why the original construction doesn't work for arbitrary $E\to M$, the derivative of the projection ${\rm Fr}(T^*M) \to M$ will take values in tangent spaces to $M$ instead of cotangent spaces (this construction should be natural and not require identifications between $TM$ and $T^*M$, and even by taking a Riemannian metric on $M$, it's not clear how this sorry attempt would carry to general $(r,s)$). I'm not sure what to try and can only guess that general bundle morphisms should enter the picture somehow.
I know that this is a bit vague but hopefully I managed to convey what I want here. Certainly someone has thought about this already, so maybe there's just some terminology I'm not aware of. Any comments are welcome. Thank you!
 A: Given a frame $f\in F_pM$, there is are corresponding $\mathbb{R}^{n^{k+l}}$-vanlued $(l,k)$ tensor $f^l_k\in T^l_kT_pM\otimes \mathbb{R}^{n^{k+l}}$ defined by
$$
\langle f^l_k,T\rangle=[T]_{\mathfrak{e}^{\otimes k}\otimes(\mathfrak{e}^*)^{\otimes l}}
$$
Where $T\in T^k_lT_pM$ is a $(k,l)$ tensor and $\langle\ ,\ \rangle$ denotes the natural pairing of $(l,k)$ and $(k,l)$ tensors. The same construction extends to a tensor field $\mathfrak{e}^l_k$ corresponding to a local frame field $\mathfrak{e}$. You are evidently looking for an object $\Theta$ such that $\mathfrak{e}^*\Theta=\mathfrak{e}^l_k$.
The purely covariant case is simple enough; for signature $(0,k)$ the $k$-th tensor power of the solder form suffices. It is a $\mathbb{R}^{n^k}$-valued $(0,k)$ tensor, so the pullback by $\mathfrak{e}$ is well defined.
In the contravariant case, however, there is no notion equivalent to a pullback unless the map is a fiberwise isomorphism, so we have to use a different bundle. Let $E^k_l=\pi^*(T^k_lTM)$ be the pullback of the tensor bundle along the frame projection $\pi:FM\to M$, and $\pi^k_l:E^k_l\to T^k_lTM$ be the lift of $\pi$. Note that $(E^k_l)^*\cong E^l_k$. We can define a section $\Theta^l_k\in\Gamma\left(E^l_k,\mathbb{R}^{n^{k+l}}\right)$ by
$$
\langle\Theta^l_k(f),T\rangle=\langle f^l_k,\pi^k_l(T)\rangle,\ \ \ T\in E^k_l|_f
$$
This object has all of the desired properties, but it is not a tensor field on $FM$ in the typical sense. There is a canonical embedding $\pi^*(T^*M)\hookrightarrow T^*FM$ which we make use of in the covariant case, but there are in general many embeddings $\pi^*(TM)\hookrightarrow TFM$ which respect pairings, and no canonical choice thereof. In fact, these embeddings are in one-to-one correspondence with Ehresmann connections on $FM$, and embeddings $\pi^*(TM)\hookrightarrow TFM$ compatible with the $GL(n,\mathbb{R})$-action are in one-to-one correspondence with principal connections. In this sense, it seems to me that some additional structure is needed if you want to define a tensor field on $FM$ itself.
