Suppose I am given two $E\_2$-ring spectra $A$ and $B$ and a morphism of $E\_2$-rings $\phi:A\to B$. Then I have $E\_1$-monoidal categories of modules $LMod\_A$ and $LMod\_B$. Moreover I have morphisms $F:LMod\_A\to LMod\_B$ and $U:LMod\_B\to LMod\_A$ where by $F$ I mean the extension of scalars functor given by $-\otimes\_A B$ and by $U$ I mean the functor that for a given $B$-module $M$ imbues it with the $A$-module structure given by $M\otimes A\overset{1\otimes\phi}\to M\otimes B\to M$. Since $LMod\_A$ and $LMod\_B$ are monoidal, I can talk about algebras therein. Is it the case that the functors $F$ and $U$ preserve this monoidal structure? If so, is there a reference for this fact within Lurie's $\infty$-categorical framework? 


I should maybe mention that Lurie remarks that the functor $F$ defined above is monoidal in Remark 6.5 of DAG VII. 



Monoidal Forgetful/Free Adjunction for $E_2$-algebras
Just so this doesn't go unanswered: the fact that $\mathrm{LMod}$ is a functor from $E\_n$-ring spectra to the $\infty$-category of $E\_{n-1}$-monoidal presentable $\infty$-categories is part of Proposition 7.1.2.6 in Higher Algebra (the other part of the proposition characterizes the image of $\mathrm{LMod}$ as consisting of those $E\_{n-1}$-monoidal presentable $\infty$-categories in which the monoidal unit is a compact generator). This tells you that the functor $F$ is $E\_{n-1}$-monoidal and a left adjoint. That $U$ is lax $E\_{n-1}$-monoidal follows from Corollary 7.3.2.7 in Higher Algebra, which basically says that the right adjoint (relative to $\mathcal{O}^\otimes$) of an $\mathcal{O}$-monoidal functor is lax $\mathcal{O}$-monoidal.


