M \ar{dd}{\psi} & \\
                    S_0\times [0,1] \sqcup \Sigmait  \ar{ru}{\alpha_- \sqcup \delta_-} \ar[swap]{dr}{\xi_0\circ \text{pr}_{S_0} \sqcup \varphi} & & S_0\times [0,1] \sqcup \Sigmait' \ar[swap]{lu}{\alpha_+ \sqcup \delta_+} \ar{dl}{\xi_1\circ \text{pr}_{S_1} \sqcup \varphi'} \\
                    & T &
                    \end{tikzcd}$$ commute. Again no compatibility on the collars is assumed.

    We define two pairs $(M,\psi)$ and $(\widetilde M,\widetilde \psi)$ to be equivalent if we can find an orientation-preserving diffeomorphism $\Phiit : M {\longrightarrow}\widetilde M$ such that the diagram $$\begin{tikzcd}
            \,        & M \ar{dd}{\Phiit} & \\
            \Sigmait \times [0,1)  \ar{ru}{\delta_-} \ar[swap]{dr}{\widetilde \delta_-} & & \Sigmait' \times (-1,0] \ar[swap]{lu}{\delta_+} \ar{dl}{\widetilde \delta_+} \\
            & \widetilde M &
            \end{tikzcd} \ ,$$ and a similar diagram involving the collars of the 1-boundary commute and if moreover $\psi = \widetilde \psi \circ \Phiit$ holds.

In order to define the vertical composition of 2-morphisms we fix a diffeomorphism $[0,2]{\longrightarrow}[0,1]$ which is equal to the identity on a neighborhood of $0$ and given by $x{\longmapsto}x-1$ in a neighborhood of $2$. Now we define the vertical composition by gluing using the collars of 0-boundaries. We use our fixed diffeomorphism to rescale both the ingoing and outgoing 1-collars.

We define horizontal composition of 2-morphisms by gluing manifolds and maps along 1-boundaries, where the new 0-collars arise from the 