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In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds X,Y and a surjective submersion f:X→Y, then f is a smoothly final map, that is for any further smooth manifold Z, and any map g:Y→Z, we have g smooth iff g∘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 states a characteristic property, but it doesn't appear on the face of it what Lee has in mind.

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# 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→Y, then f is a smoothly final map, that is for any further smooth manifold Z, and any map g:Y→Z, we have g smooth iff g∘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?