Let $R$ denote a commutative ring.
Definition. The "Leibniz multicategory" over $R$ is given as follows:
Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate').
Multiarrows. $R$-multilinear functions satisfying the "Leibniz Law." For example, if we're given an $R$-multilinear function $f : X, Y \rightarrow Z$, the Leibniz law reads:
$$Df(x,y) = f(Dx,y) + f(x,Dy)$$
The $n=3$ case is:
$$Df(x,y,z) = f(Dx,y,z) + f(x,Dy,z)+f(x,y,Dz)$$
etc.
For illustration purposes, lets show that if we're given multiarrows $$f : X,Y \rightarrow J, \quad g : J,Z \rightarrow K,$$ then the function $$h : X,Y,Z \rightarrow K \quad x,y,z \mapsto g(f(x,y),z)$$ is also a multiarrow.
We have:
$$Dh(x,y,z) = Dg(f(x,y),z) = g(Df(x,y),z)+g(f(x,y),Dz)$$
But
$$g(Df(x,y),z) = g(f(Dx,y)+f(x,Dy),z) = g(f(Dx,y),z)+g(f(x,Dy),z)$$
So
$$Dh(x,y,z) = g(f(Dx,y),z)+g(f(x,Dy),z)+g(f(x,y),Dz)$$
In other words:
$$Dh(x,y,z) = h(Dx,y,z)+h(x,Dy,z)+h(x,y,Dz)$$
as required.
Motivation. A differential $R$-algebra with one derivation is just a monoid object in the above multicategory. The obvious variant of the above multicategory to many $D's$ gives us differential algebras with many derivations.
Question. Does this multicategory (and/or its obvious generalizations) have an accepted name?
If not, is there at least somewhere I can learn more about it?