Lemma \[lemmagluing2man\] this can be verified directly.

-   Symmetric monoidal structure: There are natural equivalences of categories $$\begin{aligned}
            \iota_\theta : T_\theta (\emptyset) &\longrightarrow {{\mathbf{FinVect}_{\mathbb{C}}}}\\
            \sigma_\emptyset &{\longmapsto}{\mathbb{C}}\end{aligned}$$ and $$\begin{aligned}
            \chi_\theta((S_0, \xi_0),(S_1,\xi_1))\colon T_\theta (S_0,\xi_0)\boxtimes  T_\theta (S_1,\xi_1)  &\longrightarrow T_\theta(S_0\sqcup S_1, \xi_0 \sqcup \xi_1) \\
            \sigma_{S_0}\boxtimes \sigma_{S_1} & {\longmapsto}\sigma_{S_0} + \sigma_{S_1} ,
            \end{aligned}$$ where we suppress the inclusion into the disjoint union and denote by $\sigma_\emptyset$ the unique fundamental cycles of the empty set (that it has by convention). The modifications which are part of the structure of a symmetric monoidal 2-functor (Definition B.12 [@MS]) are trivial, since the corresponding diagrams commute on generators. The simple form of the coherence isomorphism makes it straightforward to check that the corresponding diagrams commute.

The field theory $T_\theta$ is obviously invertible.

\[Rem: Non extended theory\] Restricting $T_\theta $ to the endomorphisms of the empty set induces a non-extended homotopy quantum field theory $$\begin{aligned}
T_\theta :  T\text{-}\mathbf{Cob}(n,n-1) {\longrightarrow}{{\mathbf{FinVect}_{\mathbb{C}}}}\ ,\end{aligned}$$ which admits the following concrete description

-   To a closed $n-1$ dimensional manifold ${\Sigmait}$ equipped with a map $\varphi : {\Sigmait}{\longrightarrow}T$ it assigns the vector space $T_\theta({\Sigmait}, \varphi)={\Sigmait}^\varphi(\emptyset,\emptyset)= {\mathbb{C}}[{\operatorname{Fund}}({\Sigmait})]/{\sim}$.

-   To a morphism $(M,\psi): ({\Sigmait}_a,\varphi_a) {\longrightarrow}({\Sigmait}_b,\varphi_b)$ it assigns the linear map $$\begin{aligned}
    T_\theta(M,\psi): T_\theta({\Sigmait}_a,\varphi_a) & {\longrightarrow}T_\theta ({\Sigmait}_b,\varphi_b) \\
    [\sigma_{{\Sigmait}_a}]& 