In [Lee's Introduction to smooth manifolds](https://link.springer.com/book/10.1007/978-1-4419-9982-5) he states that given smooth manifolds $X,Y$ and a *surjective submersion*, $f:X \rightarrow Y$, then $f$ is a *smoothly final map*, that is for any further smooth manifold $Z$, and any map $g:Y\rightarrow Z$, we have $g$ smooth if $g\circ f$ is smooth.


He then says that problem 4.7 shows why this property is *'characteristic'*. I can't see why the reverse implication should hold.


Unfortunately, google-books doesn't show that page, nor do I have access to a mathematical library, can some-one enlighten me as to what he means?


One of the answers to this [question](https://mathoverflow.net/questions/115145/are-immersions-submersions-captured-in-generalised-smooth-spaces-by-some-universa) states a characteristic property, but it doesn't appear on the face of it what Lee has in mind.



What is the characteristic property of surjective submersions?
Here's what I had in mind:


**Theorem:** *Suppose $M$ and $N$ are smooth manifolds and $\pi:M\to N$ is a surjective smooth submersion. Then the given topology and smooth structure on $N$ are the only ones that satisfy the characteristic property.*


(That's what Problem 4-7 asks you to prove.)


