I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with objects all formal finite direct sums $\bigoplus\_i x\_i$ of objects $x\_i$ in $\mathcal{C}$, and morphisms
$$f\colon \bigoplus\_{i=1}^n x\_i \rightarrow \bigoplus\_{j=1}^m y\_j$$
being $m\times n$ matrices with columns indexed by $x\_i$ and rows by $y\_j$, where the $(j,i)$-th entry is a morphism $f\_{ij}\colon x\_i \rightarrow y\_j$.
Composition is given by matrix multiplication.)



> 
> What conditions (if any) must $\mathcal{C}$ satisfy in order for its
>  tensor product to lift to a tensor product in
>  $\mathcal{M}at\,\mathcal{C}$?
> 
> 
> 


Since we simply formally added the direct sums I almost feel like saying that in the case that $\mathcal{C}$ is monoidal we should just add the requirement that $\otimes$ of objects distributes over $\oplus$, then taking $\otimes$ of morphisms to be the Kronecker product of matrices (with composition in place of multiplication) seems to make everything work out.


Is it alright for me to construct a tensor product for $\mathcal{M}at\,\mathcal{C}$ from the tensor product of $\mathcal{C}$ in this way?



> 
> And is there (or is this) a general construction that makes the
>  additive envelope of a monoidal category itself monoidal?
> 
> 
> 


I have [this page](http://ncatlab.org/nlab/show/distributive+monoidal+category) off the nLab to work off, but other than that have not much else of a clue where to look, so some introductory references would also be highly appreciated. :)



Making additive envelopes of monoidal categories monoidal
Yes, your construction will work. Additive envelope is a functor 
$$\mathbf{Env} : \mathbf{Ab\text{-}Cat} \to \mathbf{Add},$$ 
from the category of Abelian enriched categories to the category of additive categories. This functor has a monoidal structure given by the obvious functors
$$\mathbf{Env}(A)\otimes \mathbf{Env}(B) \to \mathbf{Env}(A\otimes B),$$
which on objects acts as
$$\Big(\bigoplus\_ix\_i, \bigoplus\_iy\_i\Big) \mapsto \bigoplus\_{i,j}(x\_i, y\_j).$$


Thus, $\mathbf{Env}$ take monoids in $\mathbf{Ab\text{-}Cat}$, which are abilian enriched monoidal categories, to monoids in $\mathbf{Add}$, i.e. monoidal additive categories. It also follows that this construction is functorial.


