In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:


1. The sphere spectrum is the unit object
2. The tensor product bifunctor Sp x Sp -> Sp preserves colimits in each variable


Let R be an $E\_\infty$ ring. It seems clear to me like there should be a unique symmetric monoidal structure on $R$-modules such that


1. R is the unit object
2. The corresponding bifunctor preserves colimits in each variable.


but I am struggling to find a reference for this statement. Is there a standard one? Does it follow from something else in Higher Algebra?


(Even more generally, what if I have a small set of generators of a stable infinity-category such that Maps(X,Y) = 0 for any pair of distinct generators? Does a symmetric monoidal structure on the generators uniquely give a colimit-preserving symmetric monoidal structure on the entire $\infty$-category?)



Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?
If by "$R$ is the unit object" you mean : 1- the object $R$ in $Mod\_R$ is the unit **and** 2- The induced commutative algebra structure on $R = map(1,1)$ is the given commutative algebra structure on $R$, then the answer is yes.


Without the "and", namely if you only have 1-, then there are as many such symmetric monoidal structures as there are commutative algebra structures on $R$ (typically more than one).


This follows from the symmetric monoidal adjunction in *Higher Algebra*, 4.8.5.11 and 4.8.5.16 (4.8.5.11 provides the adjunction and 4.8.5.16. provides a symmetric monoidal structure on the left adjoint "$\Theta$" - $\Theta$ is the functor $R\mapsto (Mod\_R,R)$).


Together, 4.8.5.11 and 4.8.5.16 imply that the data of a colimit-preserving symmetric monoidal structure on $(Mod\_R,R)$ is equivalent to the data of an $E\_\infty$-structure on $R$, which explains my 2- above.


With regards to your parenthetical question: the property you describe provides a splitting of your category as a product of module categories (here I'm assuming you mean *compact* generators). You can then povide each factor with a symmetric monoidal structure at your leisure and get one on the product, certainly (although there might be other ones that aren't products, such as $Sp[\mathbb N]$ or something).


