Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines two subspaces of $V^{\otimes n}$, the subspace on which $S_n$ acts via the trivial character and the subspace on which $S_n$ acts via the antisymmetric character.
Question 0: Is the construction of these subspaces functorial in $V$? If it is, are the corresponding functors naturally isomorphic to the symmetric and exterior powers, and if that's true, are the corresponding natural isomorphisms unique?
If the answers to Question 0 turn out more or less like I suspect, we should not regard these subspaces as completely synonymous with the symmetric power $S^n V$ and the exterior power $\Lambda^n V$, respectively, since these are naturally thought of as quotients of $V^{\otimes n}$. (This issue recently came up in another MO question.)
Question 1: Is there an established notation in the literature which respects this distinction?
Edit: there are natural quotient maps $(V^{\ast})^{\otimes n} \to S^n(V^{\ast})$ and $(V^{\ast})^{\otimes n} \to \Lambda^n(V^{\ast})$ which dualize to inclusions $S^n(V^{\ast})^{\ast} \to V^{\otimes n}$ and $\Lambda^n(V^{\ast})^{\ast} \to V^{\otimes n}$. I guess these must be the functors I'm looking for?
