My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent"
$$B:U\to Hom(V,W)$$ where $B$ is a linear map and $$B:U\times V \to W$$
where $B$ is a bilinear map. Assuming the top, we can define our bottom map as $$(u,v)\mapsto B(u)(v)$$ and this is linear in $U$ since $B$ up top is a linear map, and linear in $V$ since $B(u)(-)$ is a linear map. Now assuming the bottom, we can just "freeze" the V component. So fix arbitrary $v\in V$, then $$B:U\to Hom(V,W)$$ $$u\mapsto B((u,v))$$
So yeah, how do I formalize this notion? And how far does this idea generalize?
$\newcommand\Vec{\operatorname{Vec}}\newcommand\Set{\operatorname{Set}}\newcommand\Hom{\operatorname{Hom}}$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction $$V\otimes_k-:\Vec_k\rightleftarrows\Vec_k:\Hom_k(V,-)$$ Thus for every vector spaces $U,W$ we have an isomorphism $$\Hom_k(V\otimes_kU,W)\cong\Hom_k(U,\Hom_k(V,W))$$ which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction $$V\times-:\Set\rightleftarrows\Set:(-)^V$$ which gives for every set $U,W$ a bijection $$\Hom(V\times U,W)\cong\Hom(U,W^V)$$ which can be written as $$W^{V\times U}\cong (W^V)^U$$