I am trying to learn some basics of category theory, precisely Adjunction, but I've encountered some difficulties trying to prove such a statement.
So, I want to prove that tensor product $- \otimes X$ and Hom-functor $\textrm{Hom}(X,-)$ form an adjoint pair. Using the definition, that means that I want to prove that $\textrm{Hom}(Y \otimes X, Z) \cong \textrm{Hom}(Y, \textrm{Hom}(X, Z))$, and iso is natural in both $Y$ and $Z$.
I am not sure if there indeed is some general proof. If there is, any help is welcome.
Something I saw as similar was that $\mathcal{L} (X; \mathcal{L}(X;Y)) \cong \mathcal{L}(X,X; Y)$, where $\mathcal{L}(X;Y)$ is space of linear maps from $X$ to $Y$. If this is such a thing, how to see the required naturality (from the definition of adjunction)? Why $\otimes$ instead of $\times$?
The following will work out the details in Brevan's comments:
I assume you are working on the category of vector spaces. Recall that by the universal property of tensor products there is a canonical (i.e., functorial with respect to all three variables) bijection $$\hom(Y\otimes X,Z)\simeq\mathrm{Bil}(Y\times X,Z),$$ where $\mathrm{Bil}(Y\times X,Z)$ is the set of bilinear maps $Y\times X\to Z$.
Moreover, there is a canonical bijection $$\begin{align*} \mathrm{Bil}(Y\times X,Z)&\simeq \hom(Y,\hom(X,Z))\\ f&\mapsto (y\mapsto (x\mapsto f(y,x)))\\ ((y,x)\mapsto g_y(x))&\leftarrow (y\mapsto g_y). \end{align*}$$ I leave it to you to check the maps are well-defined and that they are inverses of each other.