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, e.g. units.

Or do I misunderstand something?

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?

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.

Or do I misunderstand something?

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, e.g. units.

Or do I misunderstand something?