Note: I have updated my work section of this since I asked the question yesterday to reflect my understanding to this point.
Let $V$ and $W$ be two vector spaces. We denote by $S_2V$ the second symmetric power of $V$ and by $\bigwedge^2 V$ the second exterior power of $V$. Prove there exists a natural linear map $$ \phi\colon S_2(V\otimes W)\to S_2 V\otimes S_2 W $$ defined by the formula $$ \phi(v_1\otimes w_1)(v_2\otimes w_2) = (v_1 v_2\otimes w_1 w_2) $$ Show that the kernel of $\phi$ is functorially isomorphic to $\bigwedge^2 V\otimes \bigwedge^2 W$.
I think I understand how to prove the existence of a natural linear map using the universality property. If you saw my previous version, I was mistakenly trying to construct mappings from $V\otimes W$ when what is actually required is to construct the mappings from $(V\otimes W)\times(V\otimes W)$. It's fairly straightforward to set everything up and to show that the resulting map is natural.
I'm still confused about how to show that $\ker\phi$ is functorially isomorphic to $\bigwedge^2 V\otimes \bigwedge^2 W$.
I think this means that I need to construct a map $\bigwedge^2 V\otimes \bigwedge^2 W \to S_2(V\otimes W)$ and then understand what happens when I apply $\phi$. I'm not really sure about this approach, and I would appreciate any help you could offer.