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 $M_{1} ⊔ M_{2} \to \partial N$
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.