# What is the characteristic property of surjective submersions?

In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds $X,Y$ and a surjective submersion $f:X\to Y$, then $f$ is a smoothly final map, that is for any further smooth manifold $Z$, and any map $g:Y\to Z$, we have $g$ smooth iff $g\circ f$ 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, can some-one enlighten me as to what he means?

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The intention of this statement was clarified by the author Jack Lee on MathOverflow:

Here's what I had in mind:

Theorem: Suppose $M$ and $N$ are smooth manifolds and $\pi\colon M \to N$ 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.)

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Thanks. I asked the question in math overflow as I was getting any luck here... –  Mozibur Ullah Jan 27 '13 at 10:40
Sure, there's nothing wrong with that -- although next time it would be nice to provide links both here and on MO. I posted this so that you have an answer to accept and in order to let other people know that the matter was settled elsewhere. –  Martin Jan 27 '13 at 10:42
@MoziburUllah Apparently, user1421 did not notice your question here... Maybe the chance of noticing would be higher if the question was also tagged differential-geometry (manifolds by itself does not make it clear that the question is about smooth manifolds). –  user53153 Jan 27 '13 at 22:54
@5pm: you're probably right. –  Mozibur Ullah Jan 27 '13 at 23:08