Let $V$ and $W$ be vector spaces over a field $K$. Define a function $\varphi:V^*\times W\to V^*\otimes W$ as $\varphi(f, w)= f\otimes w$.
Define a function $h:V^*\times W\to \mathrm{Hom}(V, W)$ as $h(f, w)(x)=f(x)w$. By the universal property of $\varphi$, there is a natural map $\tilde{h}:V^*\otimes W\to \mathrm{Hom}(V, W)$ such that $h=\tilde{h}\circ \varphi$.
I know that if $V$ and $W$ are finite-dimensional, $\tilde{h}$ is isomorphism.
Is $\tilde{h}$ generally injective or surjective?
If so, is it can be proved without using the assumption that every vector space has a basis (axiom of choice) ?