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 $M_{1} ⊔ M_{2} \to \partial N$ is a smooth cobordism from $M_{1}$ to $M_{2}$ , and $M_{2} ⊔ M_{3} \to \partial P$ 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.