[0,1]-\sigma_1\times [0,1])\end{aligned}$$

for some fundamental cycle $\mu_b \in {\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait_b)$. Mapping $\mu_a$ to $\langle \psi^* \theta,\nu\rangle [\mu_b]$ yields a well-defined linear map $\mathbb{C}[{\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait_a)] {\longrightarrow}{\Sigmait}_b^{\varphi_b}(\sigma_1,\sigma_0)$, which descends to ${\Sigmait}_a^{\varphi_a}(\sigma_1,\sigma_0)$ and gives us the needed map .

In order to prove that $T_\theta$ strictly preserves the vertical composition of 2-morphisms, we need a Gluing Lemma for $\langle 2\rangle$-manifolds:

\[lemmagluing2man\] Given two $n$-dimensional $\langle 2 \rangle$-manifolds $M_1$ and $M_2$ with representatives for the orientation class $\nu_1$ and $\nu_2$ such that $\partial \nu_i = \mu_{i,0} + \mu_{i,1}$ for $i=1,2$, where $\mu_{i,0}$ and $\mu_{i,1}$ are representatives for the fundamental class of the 0 and 1 boundary, respectively (note that this implies $\partial \mu_{i,0}= -\partial \mu_{i,1}$). Now assume we have an orientation reversing diffeomorphism from a connected component ${\Sigmait}$ of, say, the 1 boundary of $M_1$ onto the 1 boundary of $M_2$ compatible with the fundamental cycles picked above, i.e. $\mu_{0,1}|_{{\Sigmait}}= - \mu_{1,1}|_{{\Sigmait}}$. Then $\nu_1+\nu_2$ is a representative of the fundamental class of the manifold obtained by gluing $M_1$ and $M_2$ (see figure \[Fig:Sketch Composition\]).

[Lemma28.pdf]{}(25,20)[$M_1$]{}(75,15)[$M_2$]{}(50,5)[${\color{blue}U}$]{}(50.5,14)[${\Sigmait}$]{}

We denote the composition of $M_1$ and $M_2$ by $M$. Obviously, $\nu_1+\nu_2$ is a cycle relative $\partial M$. We have to show that it represents the orientation class of $M$. To 