Cobordism of smooth maps

Definition

Suppose $M_1,M_2,P$ are differential manifolds, and $f_1:M_1\to P,f_2:M_2\to P$ are smooth maps. Then, a smooth cobordism from $f_1$ to $f_2$ is the following data:

• A smooth cobordism from $M_1$ to $M_2$ i.e. a manifold with boundary $(N,\partial N)$ with a diffeomorphism $M_1\bigsqcup M_2\to \partial N$
• A smooth map $F:N\to P$ such that the composite of $F$ with the inclusions of $M_1,M_2$ are respectively the maps $f_1,f_2$

Note that we may sometimes fix the cobordism between $M_1$ and $M_2$, and ask whether we can fill in a function $F$ for the particular cobordism.