\varphi_a : \Sigmait_b \circ \Sigmait_a{\longrightarrow}T$ is the map obtained from gluing $\varphi_a$ and $\varphi_b$. We now obtain $$\begin{aligned}
 \int^{\sigma_1 \in {\operatorname{Fund}}(S_1)} \Sigmait_a^{\varphi_a} {\left(  {\sigma_1},{\sigma_0} \right)} \otimes \Sigmait_b^{\varphi_b}{\left( {\sigma_2},{\sigma_1}\right)} &\cong \int^{\sigma_1 \in {\operatorname{Fund}}(S_1)} (\Sigmait_b \circ \Sigmait_a)^{\varphi_b \cup \varphi_a} {\left(  \sigma_2,\sigma_0\right)}  \\&\cong  \int^{\sigma_1 \in {\operatorname{Fund}}(S_1)} (\Sigmait_b \circ \Sigmait_a)^{\varphi_b \cup \varphi_a} {\left(  \sigma_2,\sigma_0\right)} \otimes \mathbb{C}  \\&\cong (\Sigmait_b \circ \Sigmait_a)^{\varphi_b \cup \varphi_a} {\left(  \sigma_2,\sigma_0\right)}, \end{aligned}$$ where in the last step we used that ${\operatorname{Fund}}(S_1)$ is connected. Insertion into yields isomorphisms $$\begin{aligned}
(T_\theta(\Sigmait_b,\varphi_b) \circ T_\theta(\Sigmait_a,\varphi_a)) \sigma_0 \cong (T_\theta( (\Sigmait_b,\varphi_b) \circ (\Sigmait_a,\varphi_a)) \sigma_0,\end{aligned}$$ which give us after linear extension the natural isomorphism .

Definition on 2-morphisms
-------------------------

To a 2-morphism $(M,\psi): (\Sigmait_a,\varphi_a) \Longrightarrow (\Sigmait_b,\varphi_b)$ between 1-morphisms $(\Sigmait_a,\varphi_a) , (\Sigmait_b,\varphi_b) : (S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ in ${T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$ we assign the 2-morphism $T_\theta(\Sigmait_a,\varphi_a) \Longrightarrow T_\theta(\Sigmait_b,\varphi_b)$ between the 1-morphisms $T_\theta(\Sigmait_a,\varphi_a), T_\theta(\Sigmait_b,\varphi_b) : T_\theta(S_0,\xi_0) {\longrightarrow}T_\theta(S_1,\xi_1)$ consisting of the natural maps $$\begin{aligned}
T_\theta(\Sigmait_a,\varphi_a) \sigma_0 {\longrightarrow}T_\theta(\Sigmait_b,\varphi_b) \sigma_0\end{aligned}$$ for $\sigma_0 \in {\operatorname{Fund}}(S_0)$ which are the maps between the respective coends induced by the linear maps $$\begin{aligned}
T_\theta(M)_{\sigma_1,\sigma_0} : {\Sigmait}_a(\sigma_1,\sigma_0)  {\longrightarrow}{\Sigmait}_b(\sigma_1,\sigma_0) \label{eqnmapbetweencoendsinduce}  \end{aligned}$$ defined as follows: For $\mu_a \in {\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait_a)$ we can find a fundamental cycle $\nu$ of $M$ with

$$\begin{aligned}
\label{EQ: Condition Fundamental cycle on manifolds with corners}
 \partial \nu = \mu_b - \mu_a +(-1)^{n-2}(\sigma_0\times 