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 $M_{1}\sqcup M_{2}\to \partial N$
• 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.