Category of vector spaces is monoidal closed

Question

How to show that the category $\textbf{Vect}_k$ of vector spaces with ground field $k$ is monoidal closed? I am aware that the right adjoint to the functor $-\otimes Y$ should be $[Y,-]: X\mapsto \textbf{Vect}_k(Y,X)$, with unit $d_X : X\to [Y,X\otimes Y]$ and counit $e_X: [Y,X]\otimes Y\to X$. What are these linear maps explicitly?

Answer 1

You don't necessarily need the unit and counit to define an adjunction: it's enough to show that the isomorphism $$[X,\, [Y,Z]]\ \cong\ [X\otimes Y,\,Z]$$ given by observing that both sides basically give the bilinear maps $X\times Y\to Z$, is natural in $X$ and $Z$.

Nevertheless, $d_X(x)=y\mapsto x\otimes y$, and
$e_X:[Y,X]\otimes Y\to X$ is $\ e_x(f\otimes y)=f(y)$.