The 357 Sig cartridge was developed by Sig in order to have a caliber that would (1) Be lighter in recoil than the 10mm, (2) Have somewhat comparable ballistics to the 357 Magnum revolver load, (3) Operate in an auto.
It is different from most auto ammo in that it is a bottle neck case (like most high powered rifle ammo). Consequently, special care must be used in reloading.
It is not really a .40 cal case that is necked down to take a 9mm bullet, but almost. The 357 sig case neck is just a little longer. There was some confusion when it first came out about it being head spaced on the rim or the shoulder. I have an earlier reloading manual that states it head spaces on the rim. However, I have since learned that it actually head spaces on the shoulder, & I do all my reloading by measuring from base to shoulder.
Ammo is quite expensive unless you reload.
The 357 Sig seems to be popular with Law Enforcement but has been slow catching on with the general shooting public.