3:10 AM
@PeterHaine OK, that's the expression I got as well. How does that impact the rest of the argument? I didn't understand exactly where it was going. It seemed more terminologically complicated than what I'd expected, which usually means there's some big point I'm missing.

3:32 AM
It doesn't affect the rest of the argument at all. The point is that I wanted to say that if M is an Aₙ- or Eₙ-algebra in spaces, then there's a natural Aₙ- or Eₙ-algebra structure on
M₊ ≔ M ⊔ {0} ,
where the multiplication on M remains the same and 0 acts as an absorbing element.
Since lax-monoidal functors preserve algebras over any operad one way to construct the desired Aₙ/Eₙ-algebra structure on M₊ is to just give the functor
(–)₊: Spc → Spc
a lax-monoidal structure.
This factors as a composite of (–)₊: Spc → Spc∗ with the forgetful functor Spc∗ → Spc. The forgetful functor Spc∗ → Spc is symmetric monoidal with respect to the product (it is a right adjoint), so it suffices to construct a lax-monoidal structure on the functor (–)₊: Spc → Spc∗ landing in pointed spaces.
The point is then that the lax-monoidal structure maps are specified by projecting from ∏ᵢ Xᵢ₊ to the summand (∏ᵢ Xᵢ)₊.
This really isn't an important point: I just wanted to construct this algebra structure S¹₊ (for the counterexample in 2.6) without referring to topological spaces or anything. But of course you can just do this by taking the actual topological monoid given by the circle plus a disjoint point that acts as an absorbing element.