In general, for infinite dimensional vector spaces, we know that $V^* \otimes W$ and $\text{Hom}(V,W)$ are not isomorphic, with $V^* \otimes W$ being isomorphic instead to the subspace of finite rank maps.
Now, suppose instead that $V$, $W$ are modules in Category $\mathcal{O}$ for some semi-simple Lie algebra (or, more generally a symmetrizable Kac-Moody algebra). If, instead of the full dual $V^* = \prod_\lambda V^*_\lambda$, we use the restricted dual $V^\vee := \bigoplus_\lambda V^*_\lambda$ (to stay inside of $\mathcal{O}$), do we recover a statement like $V^\vee \otimes W \cong \text{Hom}(V,W)$?