Category of differential manifolds with cobordisms

This article defines a category structure on manifolds (possibly with additional structure)
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Definition

The category of differential manifolds with cobordisms is defined as follows:

• The objects of the category are differential manifolds
• The morphisms of the category are smooth cobordisms i.e. the morphisms from $M_{1}$ to $M_{2}$ are the cobordisms from $M_{1}$ to $M_{2}$
• Composition of morphisms is given as follows. Suppose $M_{1}\sqcup M_{2}\to \partial N$ is a smooth cobordism from $M_{1}$ to $M_{2}$, and $M_{2}\sqcup M_{3}\to \partial P$ is a smooth cobordism from $M_{2}$ to $M_{3}$. The cobordism from $M_{1}$ to $M_{3}$ is obtained by gluing $N$ and $P$ along the image of $M_{2}$, and not changing the maps from $M_{1}$ and $M_{3}$ to the respective parts.

All the morphisms in this category are isomorphisms.

A related notion is the 2-category of differential manifolds with smooth maps and smooth cobordisms.