I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i.e. for vector spaces $U$, $V$, and $W$,
$$\mathrm{Hom}(U \otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V,W))$$
I am trying to gain some intuition for adjoints by looking at special cases. This one is puzzling me a bit. Here is one of my attempts to get a grasp of it:
If my understanding is correct, a (covariant) $k$-tensor $\omega$ on a vector space $V$ can be defined equivalently as either an element of the $k$-fold tensor product of the dual space $V^*$, i.e. $\omega \in \bigotimes_{i=1}^k V^*$, or as a $k$-form on $V$, i.e. a $k$-linear map $\omega: \bigoplus_{i=1}^k V \to \mathbb{R}$. Thus, we have a bijection between $\bigotimes_{i=1}^k V^*$ and the set of $k$-forms on $V$. Furthermore, if $V$ is finite dimensional, then we also have the well-known isomorphism $V^* \otimes V^* \cong \mathrm{Hom}(V,V^*)$.
It seems to me that these observations can be interpreted as special cases of the adjunction above, or are at least related in some way. In particular, we have
$$\mathrm{Hom}(V^* \otimes V^*, V) \cong \mathrm{Hom}(V^*, \mathrm{Hom}(V^*,V))$$
What is the proper way to interpret these facts in the language of category theory? Can this be generalized at all to help give more intuition for the adjunction in terms of the properties of tensor products?
Thank you!
There is a de-linearized version of tensor-hom adjunction, called currying in computer science, which states there is a bijection $\hom(U\times V,W)\cong \hom(U,\hom(V,W))$ in the category $\mathsf{Set}$.
Consider an arbitrary $\phi\in \hom(U\times V,W)$, i.e. a function $\phi:U\times V\to W$. By fixing $u\in U$, we can consider $\phi(u,-)$ to be a function $V\to W$. Thus the assignment $u\mapsto \phi(u,-)$ may be considered a function $U\to \hom(V,W)$, i.e. an element of $\hom(U,\hom(V,W))$. In this way, we have defined a map from $\hom(U\times V,W)$ to $\hom(U,\hom(V,W))$. I leave it as an exercise to determine how to think of the inverse map, and hence conclude we have a bijection.
There is a linearization functor $L:\mathsf{Set}\to\mathsf{Vect}$ (fix the base field $F$ to anything) which sends a set $X$ to the free vector space (over $F$) generated by $X$. I leave it as an exercise to check that $L$ is a monoidal functor, i.e. figure out how functions $X\to Y$ turns into linear maps $LX\to LY$ and how $L(X\times Y)\cong L(X)\otimes L(Y)$. This suggests tensor-hom adjunction comes from applying linearization to the currying adjunction.
Indeed, starting with $\phi\in \hom(U\otimes V,W)$, i.e. a bilinear map $\phi:U\times V\to W$, we can fix a vector $u\in U$ and consider the linear map $\phi(u,-):V\to W$. The assignment $u\mapsto \phi(u,-)$ is a function $U\to\hom(V,W)$ which we can check is linear, so an element of $\hom(U,\hom(V,W))$, and again we can figure out the inverse and hence conclude this is an isomorphism.
Homs between vector spaces can be related to tensor products and dual spaces as well. Indeed there is a natural isomorphism $V^*\otimes W\cong \hom(V,W)$. (One must consider $\mathsf{Vect}$ to be "enriched" so that $\hom(-,-)$ is a $\mathsf{Vect}$-valued bifunctor.) Given a pure tensor $\phi\otimes w\in V^*\otimes W$, it can be reinterpreted as a linear map $V\to W$ defined by $(\phi\otimes w)(v):=\phi(v)w$. By picking bases for $V$ and $W$ we can prove this process of reinterpretation induces an isomorphism between $V^*\otimes W$ and $\hom(V,W)$. When combined with $(V^*)^*\cong V$ and $(V\otimes W)^*\cong V^*\otimes W^*$, this gives us a lot of power when it comes to manipulating tensors and homs of spaces.
I'll also point out there are "diagrammatic" interpretations of these operations between spaces and their manipulations and natural isomorphisms. See Qiaochu's two-part blog posts on string diagrams (Part I, Part II).