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?

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?