Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:



> 
> In many cases, the category $\cal{M}od\_A\left(\cal{C}\right)$ of $A$-modules in
>  $\cal{C}$ inherits the structure of a symmetric monoidal category with respect to the relative tensor product over $A$.
> 
> 
> 


Where can I find conditions, details and proofs of this (seemingly) elementary fact? (I didn't find it in the book I searched - MacLane, Barr & Wells, or in books about operads etc.)


I also need the facts that "extension of scalars" $-\otimes A$ is the left adjoint of the forgetful functor and it commutes with the tensor product. And the case when $\otimes$ distriute over the coproduct (or the bi-product).


I think I can prove most of these facts, but I want to be sure about them, and it would also be much easier to refer to them.



Category of modules over commutative monoid in symmetric monoidal category
One needs that $C$ is cocomplete and that $\otimes$ preserves colimits in each variable (and then $\mathrm{Mod}\_C(A)$ will have the corresponding properties). More precisely, you only need that $C$ has reflexive coequalizers and that $\otimes$ preserves them in each variable. This has been known for decades, but the first clear write-up of this, at least I know of, is [Florian Marty's](http://thesesups.ups-tlse.fr/540/) thesis. You can also find a discussion on this in my [thesis](http://arxiv.org/abs/1410.1716), Section 4.1 (and Chapter 6 for the issue on reflexive coequalizers). See also [MO/114457](http://athoverflow.net/questions/114457) for a discussion of the internal homs by Todd Trimble.


*Edit.* Here are some other references, which even discuss the case where $- \otimes A$ is replaced by a (suitable) symmetric monoidal monad:



> 
> H. Lindner. Commutative monads. In Deuxiéme colloque sur
>  l'algébre des catégories. Amiens-1975. Résumés des conférences,
>  pages 283-288. Cahiers de topologie et géométrie différentielle
>  catégoriques, tome 16, nr. 3, 1975.
> 
> 
> R. Guitart. Tenseurs et machines. Cahiers de Topologie et
>  Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.
> 
> 
> A. Kock. Closed categories generated by commutative monads.
>  Journal of the Australian Mathematical Society, 12(04):405-424,
>  1971.
> 
> 
> G. J. Seal. Tensors, monads and actions. Theory Appl. Categ.,
>  28:No. 15, 403-433, 2013.
> 
> 
> 


If I recall correctly, some of these references restrict the monads in such a way that they are of the form $- \otimes A$ anyway.


