The algebra of natural transformations of the n-th power tensor functor

Question

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group.

Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote $Vect_k$. Let $T^{(n)}$ be the functor $[V \mapsto V^{\otimes n}]$ where $V^{\otimes n}$ is the $n$-th tensor power of $V$ and $[f\mapsto f^{\otimes n}]$ where $f^{\otimes n}$ is the $n$-th tensor power of $f$. I know there is a faithfull action of $S_n$ on $T^{(n)}$ i.e. there is an injection $S_n \hookrightarrow Nat(T^{(n)}, T^{(n)})$ where $Nat(T^{(n)},T^{(n)})$ is $k$-algebra of natural transformations form $T^{(n)}$ to its self. Because of this I have an injection of the algebra of the symmetric group in $Nat(T^{(n)},T^{(n)})$ i.e. $$ k[S_n] \hookrightarrow Nat(T^{(n)}, T^{(n)}).$$

My questions are :

1. Is this injection an isomorphism of $k$-algebras? or are there other natural transformations from $T^{(n)}$ to its self?
2. Does it have to do with the $Vect_k$ category or is it something more general about symmetric monoidal categories?
3. If the category is just braided can we say the same kind of things for $k[B_n]$ where $B_n$ is the braided group on $n$ strings?

I came across this by trying to prove that $k[S_n]$-modules are the same as homogeneous polynomial functors of degree $n$.

Answer 1

1. Yes. This is a corollary of Schur-Weyl duality.

2. You need at least the additional assumption that your symmetric monoidal category is enriched over $k$-vector spaces. In general I don't see any reason to expect that the action of $k[S_n]$ is faithful; consider, for example, the special case where we only look at $1$-dimensional vector spaces.

3. Sometimes. The keyword to look up here is "quantum Schur-Weyl duality."