The 9mm barrel will not fit in the 40 frame without narrowing at the base.
(Then it will fit.)
The breech face on the 40 slide is larger and therefore the extractor is farther from the case rim.
The 40 barrel bushing is also larger than the 9mm.
Also the 40 recoil spring is heavier than the 9mm.
The 9mm mag will lock in the 40 magwell (but not vice-versa).
The conversion I've seen utilized a thinned 9mm barrel with 9mm slide and recoil spring with a 9mm mag.
The TSW models are a different story for a different time.
John