\[defmscbordcattarget\] For $n\ge 2$ and any non-empty topological space $T$, which we will refer to as the *target space*, we define the bicategory $T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)$ as follows:

1.  Objects are pairs $(S,\xi)$ consisting of a $n-2$-dimensional oriented closed manifold $S$ and a continuous map $\xi: S {\longrightarrow}T$ (hereafter just referred to as map).

2.  A 1-morphism $(\Sigmait,\varphi) : (S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ is an oriented compact collared bordism $(\Sigmait,\chi_-,\chi_+) : S_0 {\longrightarrow}S_1$ (by this we mean a compact oriented $n-1$-dimensional manifold $\Sigmait$ with boundary equipped with orientation preserving diffeomorphisms $\chi_-\colon S_0\times [0,1) {\longrightarrow}\Sigmait_-$ and $\chi_+:  S_0\times (-1,0] {\longrightarrow}\Sigmait_+$ with $\Sigmait_- \cup \Sigmait_+$ being a collar of $\partial \Sigmait$), and a map $\varphi : \Sigmait {\longrightarrow}T$ making the diagram $$\begin{tikzcd}
            \, & \Sigmait \arrow{dd}{\varphi}  & \\
            S_0\times\{0\}  \arrow{ru}{\chi_-} \arrow[swap]{dr}{\xi_0\circ \operatorname{pr}_{S_0}} & & S_1\times \{0\}  \arrow[swap]{lu}{\chi_+} \arrow{dl}{\xi_1\circ \operatorname{pr}_{S_1}} \\
            & T  &
            \end{tikzcd}$$ commute. No compatibility on the collars is assumed. We define composition of 1-morphisms by gluing of bordisms along collars and maps. The collars are needed to define a smooth structure on the composition. The identities are cylinders equipped with the trivial homotopy, where trivial means constant along the cylinder axis.

    [2-Morphisms.pdf]{}(79,53)[$T$]{}(40,30)[$M$]{}(40,0)[${\Sigmait}_a$]{} (34,45)[${\Sigmait}_b$]{} (80,10)[$\varphi_a$]{}(40,51)[$\varphi_b$]{} (60,45)[$\psi$]{}(0,35)[$\xi_0 \times [0,1]$]{} (68,30)[$\xi_1 \times [0,1]$]{}

3.  A 2-morphism 