How to prove that $V \otimes W \cong V^{\ast \ast} \otimes W$? Suppose $V$ is a vector space of finite dimension. We know that $V$ and $V^{\ast \ast}$ are canonically isomorphic via the isomorphism $v \mapsto E_v$, with $E_v: V^\ast \to \mathbb{C}: f \mapsto f(v)$. 
Now I want to prove that $V \otimes W \cong V^{\ast \ast} \otimes W$, what is the easiest way to do this? 
My initial idea: define a bilinear map $\phi: V \times W \to V^{\ast \ast}  \otimes W: (v,w) \mapsto E_v \otimes w$. This is bilinear since $\otimes$ is bilinear and $v \mapsto E_v$ is linear. Thus, by the universality property this gives us a unique linear map $\overline{\phi}: V \otimes W \to V^{\ast \ast} \otimes W$ with $\overline{\phi}(v \otimes w) = E_v \otimes w$. 
But how can we show that this is an isomorphism? We only know it is linear, what about bijectivity? I don't know how to go about showing that, since the tensor product is defined indirectly. 
 A: Hint: What does it mean to be in the kernel of this map? What element maps to $E_v \otimes w$ (which is an arbitrary generating element)?
Elaboration
Suppose $\bar{\phi}(v \otimes w) = 0_{V^** \otimes W}$. What can we say about $v \otimes w$? You could also consider two simple tensors $v\ otimes w$ and $v' \otimes w'$ which map to the same place.
$$\bar{\phi}(v \otimes w)=\bar{\phi}(v' \otimes w')$$
Let $E_v \otimes w \in V^** \otimes W$. Can you find an element that maps to it? (like $v \otimes w$).
A: This works more generally. Let $V\cong U$ with isomorphisms $\varphi\colon V\to U$ and $\psi\colon U\to V$ being inverse to each other.
Similar to what you've done, you can now induce maps $$\varphi\otimes \operatorname{id}_W\colon V\otimes W\to U\otimes W,\\ v\otimes w\mapsto \varphi(v)\otimes w,$$ and $$\psi\otimes \operatorname{id}_W\colon U\otimes W\to V\otimes W,\\
u\otimes w\mapsto \psi(u)\otimes w.$$
So, the question is, what are the compositions $(\varphi\otimes \operatorname{id}_W)\circ(\psi\otimes \operatorname{id}_W)$ and $(\psi\otimes \operatorname{id}_W)\circ(\varphi\otimes \operatorname{id}_W)$?
