I'm studiyng the Schur Weyl duality in this PDF (http://www.math.columbia.edu/~ums/Finite%20Group%20Rep%20Theory7.pdf) and i'm trying to understand the following theorem
Theorem: The subalgebra of $End(V^{\otimes n})$ spanned by the image of $\mathfrak{gl}(V)$ is $B = End_{\mathbb{C}[S_{n}]}(V^{\otimes n})$
In the proof the author states: $$B = End_{\mathbb{C}[S_{n}]}(V^{\otimes n})$$ $$= End(V^{\otimes n})^{S_{n}}$$ $$\cong(End(V^{\otimes n}))^{S_{n}}$$ $$\cong Sym^{n}End(V)$$
I would like to know the definition of $End(V^{\otimes n})^{S_{n}}$. Also, I would like to understand how to prove this sequence of isomorphisms.
Thank you in advance