Let $R$ be a commutative ring and $A_1,...,A_{n+1}$ be $R$-algebras.
Let $A_1\otimes_R\cdots\otimes_R A_{n+1}$ be equipped with the natural $R$-algebra structure.
Let $N$ be an $R$-algebra.
Let $\phi:A_1\times\cdots \times A_{n+1}\rightarrow N$ be a middle linear map such that $\phi(a_1b_1,...,a_{n+1}b_{n+1})=\phi(a_1,...,a_{n+1})\phi(b_1,...,b_{n+1})$
Then, there exists a unique $(R,R)$-bimodule & $R$-algebra homomorphism $\Phi:A_1\otimes_R ...\otimes_R A_{n+1}\rightarrow N$ such that $\Phi(a_1\otimes...\otimes a_{n+1})=\phi(a_1,...,a_{n+1})$.
Is there a name for maps such as $\phi$? $R$-multialgebra?