In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given $M,N\in\mathrm{Cob}(n)$ a morphism $M\to N$ is an $n$-dimensional manifold $B$ equipped with an orientation preserving diffeomorphism $\partial B\simeq \overline{M}\coprod N$ where $\overline{M}$ denotes the manifold $M$ equipped with the opposite orientation.
What is the necessity of having one part of the boundary have the reverse orientation? If we define it so that the above equation is simply $M\coprod N$, do we run into problems?
Thanks!