Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\in M$ and every non-negative integers $k$. Defininig relations between the generators are given by


$m^{(0)} = e$


$m^{(k)}m^{(l)} = \binom{k+l}{l}m^{(k+l)}$


$(m\_1+m\_2)^{(l)} = \sum\_{k=0}^l m\_1^{(k)}m\_2^{(l-k)}$


$(am)^{(k)}=a^k m^{(k)}$.


The algebra $D(M)$ is graded with the degree function given by $deg(m^{(k)})= k$.


Then $D$ is a functor from the category of $R$-modules to the category of graded $R$-algebras in the obvious way. By passing to the $r$-homogeneous component of $D(M)$ we get the endofunctor $D\_r$ on the category of $R$-modules. 


Then it seems functor $D\_r$ can be given a structure of a weakly monoidal functor as follows. 


The unit transformation $\eta\colon R\to D\_r(R)$ is defined by $\eta(a)=a^re^{(r)}$.


The multiplication transformation $\tau\colon D\_r(M)\otimes D\_r(N)\to D\_r(M\otimes N)$ is defined by 


$$m\_1^{(k\_1)}\dots m\_t^{(k\_t)} \otimes n\_1^{(l\_1)}\dots n\_s^{(l\_s)} \mapsto \sum\_{v} \prod\_{i=1}^t \prod\_{j=1}^s (m\_i\otimes n\_j)^{(v\_{ij})},$$


where summation is over matrices $v\in M\_{t,s}(\mathbb{N}\_0)$ such that the sum of rows of $v$ is $(l\_1,\dots,l\_s)$ and the sum of collumns of $v$ is $(k\_1,\dots,k\_t)$. 


Is there any reference where such monoidal structure on $D\_r$ is considered?



Is the functor of divided powers a weakly monoidal functor? 
This is a well-known result and, apart from terminology, should be found in
Roby, Norbert
Lois polynômes multiplicatives universelles. (French. English summary)
C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 19, A869–A871. 


Note that this becomes most natural if one interprets your $D\_r(M)$ (usually denoted $\Gamma^r(M)$ and the map $M\to \Gamma^r(M)$ (given by $m\mapsto m^{(r)}$ as the universal homogeneous polynomial map of degree $n$. From this it is quite formal to conclude that there is a map $\Gamma^r(M)\bigotimes \Gamma^r(N)\to \Gamma^r(M\bigotimes N)$ characterised by $m^{(r)}\otimes n^{(r)}\mapsto (m\otimes n)^{(r)}$ (and commutation with scalar extension).


**Addendum**: I don't have access currently to Roby's article so let me give the argument.
We have a map $M\times N\to \Gamma^r(M\bigotimes N)$ given by $(m,n) \mapsto (m\otimes n)^{(r)}$. This commutes with extension of scalars and is hence a polynomial map in Roby's sense. It is also bihomogeneous of degree $r$ and hence gives a linear map $\Gamma^r(M)\bigotimes \Gamma^r(N)\to \Gamma^r(M\bigotimes N)$. 


Applied to the case when $A=M=N$, where $A$ is an algebra we get a map $\Gamma^r(A)\bigotimes \Gamma^r(A)\to \Gamma^r(A\bigotimes A)$ which composed with the multiplication map $A\bigotimes A\to A$ gives the algebra structure that Roby is considering.


