TITLE: How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?
QUESTION [9 upvotes]: Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving morphisms between these two categories. How do we characterize them? If $n=\infty$ we just have maps from $A$ to $B$ and if we don't ask for any monoidal compatibility I guess we just get bimodules over $A$ and $B$, but what are the intermediates? Maybe we can also ask for a characterization of $E_n$ maps of the category of $E_n$ modules over $E_n$ rings?
Edit: Thanks to Dylan's answer below the first question is solved. However I believe the second question is still not answered, which is about studying $E_n$ maps between categories of $E_n$ modules over $E_n$ rings.

REPLY [2 votes]: To clarify my comment: suppose you're interested in lax $E_n$-monoidal functor (so that indeed, you get coherent maps $F(M)\otimes F(N)\to F(M\otimes N)$ which aren't necessarily equivalences, similarly for $1\to F(1)$) that preserve colimits, then the category of such functors is essentially $Alg_{E_n}(Fun^{ex}(Mod_A^\omega,Mod_B))$ where $Fun^{ex}(Mod_A^\omega, Mod_B)$ has a symmetric monoidal structure induced by the Day convolution structure on $Fun(Mod_A^\omega, Mod_B)$.
Then you get a symmetric monoidal identification $Mod_{A^{op}\otimes B} \simeq Fun(Mod_A^\omega,Mod_B)$ which, since $A$ is $E_\infty$, can be thought of as $Mod_{A\otimes B}$.
Therefore lax $E_n$-monoidal colimit preserving functors $Mod_A\to Mod_B$ can be thought of as $E_n-A\otimes B$-algebras.
Of course, these rarely classify strict monoidal functors (those where $F(M)\otimes F(N)\to F(M\otimes N)$ is an equivalence, and $1\to F(1)$ is as well). In fact they do if and only if this $A\otimes B$-module is $B$ with structure induced from an $E_{n+1}$-map $A\to B$: Tyler's answer shows this.
I made a silly comment below Tyler's answer (then deleted, as I realized my mistake), which is perhaps noteworthy: if you try to take this approach and check when, for an $A\otimes B$-module $R$, the associated lax $E_n$-monoidal functor is actually strict, you'll get two conditions: one on the map $F(M)\otimes F(N)\to F(M\otimes N)$ and one on the unit. The second one is easy to forget (which is what happened in my silly comment), and if you do, you'll get something like an $E_n-B$-algebra such that the structure map $B\to R$ is an epimorphism.
This is why for instance $Mod_\mathbb Q\to Sp$ is almost symmetric monoidal, although it is only lax: you get the condition on tensor products but not the one on the unit.