the notation $$\begin{aligned}
{\Sigmait}^\varphi {\left( \sigma_1, \sigma_0\right)} := \frac{\mathbb{C}[{\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait)]}{\sim}\end{aligned}$$ for the quotient and observe that ${\Sigmait}{\left( ?,\sigma_0\right)}$ extends to a functor ${\operatorname{Fund}}^{{\text{opp}}}(S_1){\longrightarrow}{{\mathbf{FinVect}_{\mathbb{C}}}}$, which is defined on a morphism $\lambda : \sigma_1 {\longrightarrow}\sigma_2$ in ${\operatorname{Fund}}(S_1)$ by $$\begin{aligned}
{\Sigmait}{\left( ?,\sigma_0\right)} (\lambda):  {\Sigmait}{\left( \sigma_2,\sigma_0\right)} & {\longrightarrow}{\Sigmait}{\left( \sigma_1,\sigma_0\right)}   \\
\mu &{\longmapsto}\mu- \lambda \ .\end{aligned}$$ A straightforward calculation shows that this is well-defined.

Now $T_\theta(\Sigmait,\varphi) : T_\theta (S_0,\xi_0) {\longrightarrow}T_\theta(S_1,\xi_1)$ is defined on objects by the coend $$\begin{aligned}
T_\theta(\Sigmait,\varphi) \sigma_0 := \int^{\sigma_1 \in {\operatorname{Fund}}(S_1)} \Sigmait {\left( \sigma_1,\sigma_0\right)} * \sigma_1 \end{aligned}$$ and linear extension. The coend can be replaced by an end, since it is taken over a groupoid and limits and colimits over groupoids taken in a 2-vector space coincide.

Before completing the definition of $T_\theta$ we already show that it respects the composition of 1-morphisms up to coherent natural isomorphism. This is a crucial part of the 2-functoriality of $T_\theta$. For the proof we will need the following Gluing Lemma. It is a special case of the Gluing Lemma for $\langle 2\rangle$-manifolds, which will appear as Lemma \[lemmagluing2man\] below, hence we omit the proof. In the statement we suppress the inclusions of boundary components in the notation.

\[lemmamscgluingcyclescob\] Let $\Sigmait_a \colon S_0 {\longrightarrow}S_1$ and $\Sigmait_b 