Background: I am refreshing my knowledge of tensor dualities to catch up with some physical applications.
Example 1: I am aware that $\mbox{Hom}(V, \mbox{Hom}(W,U))$ is naturally isomorphic to $\mbox{Hom} (V\otimes W, U)$ and a linear $f\colon V\to \mbox{Hom}(W,U)$ is in a natural way identical to a linear $g\colon V\otimes W \to U$.
Example 2: I understand the applications of this for multilinear forms. For example, there is natural isomorphism between linear functions $f\colon V\otimes W \to {\mathbb K}$ and linear functions $g\colon V \to W^*$. Using a symmetric natural transformation, this also is naturally isomorphic to linear functions of the type $h\colon W \to V^*$.
Approaching my question: Now I became curious. The situation looks to me that there is some general theorem for moving back and forth components of a tensor products left and right of the linear mapping arrow $\to$ or, equivalently, $\mbox{Hom}$ functor in the vategory of vector spaces (or: finite vector spaces).
Conjecture 1: I can push a left-side tensor factor to the right-side by adding a dualization and I probably get a naturally isomorphic space. More precisely: The space of functions of type $V_1 \otimes \ldots \otimes V_k \to W_1 \otimes \ldots \otimes W_l$ is naturally isomorphic to the space of functions of type $V_2 \otimes \ldots \otimes V_k \to V_1^* \otimes W_1 \otimes \ldots \otimes W_l$
Conjecture 2: I can pull a right-side tensor factor to the left-side by deleting (or, maybe, also adding? not sure?!?) dualization. Removing an existing dualization star should work. Adding a further dualization star should(?) work as well, at least if the component is finite dimensional, due to the natural isomorphism $V \cong V^{**}$ for finite dimensional vector spaces.
Question: To me it looks like there is a more general theorem for moving tensor product components (with or without dualization star) from one side of the $\mbox{Hom}$ to the other side and staying naturally isomorphic while doing so.
What is known there? Is there a more general theorem? How does it look like?
And if there is: Is there, among all of these equivalent, naturally isomorphic forms, some form of generic or normal form?
I am interested in the finite as well as the infinite dimensional case. I am focusing on naturally isomorphic scenarios.
For finite dimensional spaces, we have canonical isomorphism $\newcommand{\Hom}{\operatorname{Hom}}$ : \begin{align} \Hom(U,V) &\simeq U^*\otimes V \\ (U\otimes V)^* &\simeq U^*\otimes V^* \\ U^{**}&\simeq U \end{align} as well as canonical isomorphism for the associativity and commutativity of the tensor product.
From those we can derive other isomorphisms : \begin{align} \Hom(U\otimes V,W) &\simeq (U\otimes V)^*\otimes W\\ &\simeq U^*\otimes V^*\otimes W \\ &\simeq \Hom(U,V^*\otimes W) \end{align}
Likewise : \begin{align} \Hom(U,V\otimes W) &\simeq U^*\otimes V\otimes W\\ &\simeq (U\otimes V^*)^*\otimes W \\ &\simeq \Hom(U\otimes V^*,W) \end{align} and : \begin{align} \Hom(U,\Hom(V,W)) &\simeq \Hom(U,V^*\otimes W) \\ &\simeq \Hom(U\otimes V,W) \end{align}
From this set of canonical isomorphims, you can build all the canonical isomorphisms you want. I am not sure what kind of theorem you want to state/prove here.
Using those isomorphisms, we can put a space of the form : $$\mathfrak V = \Hom(U_1 \otimes \ldots\otimes U_i \otimes V_1^* \otimes \ldots \otimes V_j^*,W_1\otimes \ldots\otimes W_k\otimes E_1^* \otimes \ldots \otimes E_n^*)$$ into any number of "normal" forms which are unique up to permutation of the factors :