Gamma spaces and monoidal categories II This question is kind of a follow-up of this one. 
Suppose I have a topological category $\mathcal{C}$ (objects and morphisms topological spaces, source and target map continuous, etc.) together with a continuous tensor product $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, such that it is strict monoidal and symmetric, but there is no unit object (I have some kind of homotopy unit, in the cases I am interested in, but I don't know how to build it into the category). 


What is the structure of the classifying space $B\mathcal{C}$? Does the Gamma-space construction of Segal still work and give me some kind of homotopy associative $H$-space?


 A: I think you do not need to use the Gamma space construction of Segal here. You just need to note that the classyfing space functor from topological categories into toopological spaces preserves products (given you work with compactly generated spaces).
This implies that in your situation the classifying spaces $|C|$ inhertits a strict multiplication, i.e. it is a topological monoid (possibly without unit). The homotopy unit on the topoloigal category will then lead to a homotopy unit for your monoid. 
If you now want to group complete you form $\Omega B |C|$. Note here that $B|C|$ can be formed by taking the fat geometric realization of the simplicial space $N|C|_n := |C|^{n-1}$. For this you don't need degeneracies, i.e. units.  
Or do I misunderstand something?
A: You can construct a $\Gamma_{epi}$-category from your category (where $\Gamma_{epi}$ denotes the category of finite pointed sets with epimorphisms). You cannot extend it to a $\Gamma$-category if your symmetric monoidal category doesn't have a unit (if you could, this would give you a unit). When you apply nerve you get a $\Gamma_{epi}$-space and this is probably the same as a non unital $E_{\infty}$-space (i.e. a space over the operad $E_{\infty}$ where you forgot about the $0$-th space). 
However there is a result of Lurie in Higher Algebra that says that homotopy units on non unital $E_{\infty}$-spaces can be strictified so that the result is a unital $E_{\infty}$-space. You seem to have a homotopy unit in your category so this result might be relevant.
