Is there a relation between $End(M)$ and $M$ under tensor products? Let $R$ be a commutative ring and $M$ be an $R$-module (If necessary, you can assume more conditions)
Let $\phi$ be an $R$-endomorphism on $M$ and $\overline{\phi}:M^{\otimes n}\rightarrow M^{\otimes n}$ be the tensor product of $\phi$. (I used the notation $\overline{\phi}$ to distinguish the one that will come below)
Now let consider the tensor power of $End(M)$. That is, $End(M)^{\otimes n}$. And consider a tensor of it, namely, $\phi^{\otimes n}$.
Is there a relation between $\overline{\phi}$ and $\phi^{\otimes n}$?
I think the reason why I'm asking this is I don't actually have a picture of the concept tensor..
 A: Consider the case $n=2$. The only general answer I know is this: by the universal property of  tensor products, given $R$-modules $M_1,M_2,N_1, N_2$ there is a canonical homomorphism:
$$\DeclareMathOperator{\Hom}{Hom}\Hom_R(M_1,N_1)\otimes_R\Hom_R(M_2,N_2)\to\Hom_R(M_1\otimes_RM_2, N_1\otimes_R N_2)$$
and this homomorphism is an isomorphism if one of the pairs $(M_1,M_2)$, $(M_1,N_1)$, $(M_2,N_2)$ consists in finitely generated projective $R$-modules (Bourbaki, Algebra, Ch. II, Linear Algebra, §4 no4, prop.4).
 So in your the answer is they're isomorphic for finite projective $R$-modules.
Note:  This covers the case of finite dimensional vector spaces.
A: $\require{enclose}$
I just reformulate the rusults form vector spaces that i know.
For any modules $M_1,N_1,M_2,N_2$ there is an $\enclose{horizontalstrike}{\text{injective}}$ (EDIT. Apparently injectivity doesn't hold for modules over rings. Modules require additional conditions.) mapping $$j:Hom(M_1,N_1)\otimes Hom(M_2,N_2)\rightarrow Hom(M_1\otimes M_2,N_1\otimes N_2),$$ 
such that $$j(\phi\otimes\psi)(m_1\otimes m_2)=\phi(m_1)\otimes \psi (m_2).$$
So we can identify elements in $Hom(M_1,N_1)\otimes Hom(M_2,N_2)$ with elements in $Hom(M_1\otimes M_2,N_1\otimes N_2)$ (if this additional conditions hold). Hence given simple $\phi\in End(M)^{\otimes n}$ we can think of this as an element of $End(M^{\otimes n}).$ So
$$\phi^{\otimes n}(m_1\otimes\dots\otimes m_n)=\phi(m_1)\otimes\dots\otimes \phi(m_n)=\bar\phi(m_1\otimes\dots\otimes m_n).$$

If you want to read more, here is the reference:
Greub, Multilinear Algebra, Springer 1967.
