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 forgetful functor $U$ defined above is monoidal in Remark 6.5 of DAG VII.

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.

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?

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 forgetful functor $U$ defined above is monoidal in Remark 6.5 of DAG VII.