Lee's Introduction to Smooth Manifolds includes the following theorem (Thm. 4.30):
Suppose $M$ and $N$ are smooth manifolds and $\pi : M \to N$ is a surjective smooth submersion. If $P$ is a smooth manifold with or without boundary and $F : M \to P$ is a smooth map that is constant on the fibers of $\pi$, then there exists a unique smooth map $\tilde{F} : N \to P$ such that $\tilde{F} \circ \pi = F$.
He proves this by using the related result for topological spaces and quotient maps to construct $\tilde{F}$ uniquely as a continuous map and then verifies that it is smooth by a characteristic property (Thm. 4.29). But this seems like overkill. Wouldn't it suffice to construct $\tilde{F}$ uniquely as a simple set map, without involving the topology of quotient spaces at all?