So, a lax $E_n$-monoidal map $BG\to Spectra$ picks out a $G$-action on an $E_n$-algebra, by $E_n$-algebra maps. I want to know if I can compute the colimit of this diagram in the underlying category of spectra.
But... okay maybe I'm mixing things up here.
That doesn't really bring $BG^\otimes$ in at all.
Yeah, hrm. I'm not seeing now why I was worried about that.
So, let's say I have a lax $E_n$ map $BG\to Spectra$ picking out a $G$-action on an $E_n$-algebra $X$. Then I want to know whether or not this functor actually factors through $BGL_1(X)$, I guess...
which would require the maps going to $X$-module morphisms.
We know that they go to $S$-algebra morphisms.
Which means they must be compatible with $X$'s action on itself.
So they must be $X$-module morphisms w/r/t $X$'s action on itself.
So this functor can be lifted to $BGL_1(X)$ and then, by inclusion, to $LMod^{E_n}_X$.
The question I was bothering with was whether or not the colimit computed in $LMod_X^{E_n}$ was the same as the colimit computed in $Spectra$
And this seems to depend on whether or not $BG$ is sifted, but NOT on whether or not $BG^\otimes$ is sifted.
@SaulGlasman when doing operadic stuff, it may also be useful to know that the category $Disk(n)$ (of Ayala, Francis etc.) is sifted. I don't know if this means that something in the shape of $Disk(n)$ (e.g. an $E_n$-algebra) is sifted... but it seems possible?
But I don't think $BG$ is sifted...