$(\Sigmait,\varphi) \Longrightarrow (\Sigmait',\varphi')$ between 1-morphisms $(S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ is defined to be an equivalence class of pairs $(M,\psi)$ consisting of an $n$-dimensional collared compact oriented bordism $M : \Sigmait {\longrightarrow}\Sigmait'$ with corners and a map $\psi : M {\longrightarrow}T$ (see figure \[Fig:Sketch 2-Morphism\]). Recall that an $n$-dimensional collared compact oriented bordism is a $\langle 2 \rangle$-manifold $M$ equipped with

    -   a decomposition of the 0-boundary $\partial_0 M = \partial_0 M_- \cup \partial_0 M_+$ and orientation preserving diffeomorphisms $\delta_- : \Sigmait \times [0,1) {\longrightarrow}M_-$ and $\delta_+ : \Sigmait' \times (-1,0] {\longrightarrow}M_+$ to collars of this decomposition,

    -   a decomposition of the 1-boundary $\partial_1 M = \partial_1 M_- \cup \partial_1 M_+$ and orientation preserving diffeomorphisms $\alpha_- : S_0 \times [0,1)\times [0,1] {\longrightarrow}N_-$ and $\alpha_+ : S_1 \times (-1,0]\times [0,1] {\longrightarrow}N_-$ to collars of this decomposition making for some $ \varepsilon >0$ the diagrams $$\label{Condition Collars 1}
                    \begin{tikzcd}
                    S_0 \times [0,1)\times [0,\varepsilon) \ar{r}{\alpha_-}  \ar[swap]{rd}{\chi_-\times {\text{id}}}& M & S_1 \times (-1,0]\times [0,\varepsilon) \ar[swap]{l}{\alpha_+} \ar{ld}{\chi_+\times {\text{id}}} \\
                    & \Sigmait \times [0,\varepsilon) \ar[swap]{u}{\delta_-}&
                    \end{tikzcd},$$ $$\label{Condition Collars 2}
                    \begin{tikzcd}
                    \,          & \Sigmait' \times (-\varepsilon,0] \ar{d}{\delta_+}   & \\
                    S_0 \times [0,1)\times (1-\varepsilon,1] \ar{r}{\alpha_-} \ar{ru}{\chi'_-\times {\text{id}}-1}& M & S_1 \times (-1,0]\times (1-\varepsilon,1] \ar[swap]{l}{\alpha_+} \ar[swap]{lu}{\chi'_+\times {\text{id}}-1}
                    \end{tikzcd}$$ and $$\begin{tikzcd}
                    \,        & 