Correct. It was Smith & Wesson's second revolver which was produced from 1861 until 1874 and was chambered in .32 rimfire. While never purchased by the Union war department, thousands of the were purchased by Union soldiers for self defence due to the fact they were breechloaders that shot a self contained metallic cartridge during a war that was mostly fought with muzzle loading firearms.
Here's a tintype of a Union cavalryman showing off his No. 2 along with having a conventional 1858 Remington cap and ball revolver tucked in his belt.
General Rutherford B. Hayes (later our 19th president) favored the little model 2.
The pistol was favored by George Armstrong Custer who was presented a cased and engraved set after the ACW.
James Butler Hickok, aka, Wild Bill Hickok occasionally carried a model 2. Hickok and Custer were well acquainted with each other--since Hickok had scouted for Custer at some point.
Of course Custer went to meet his fate at the Little Bighorn in June of 1876. Two months later in August, Hickok was shot in the back of the head while playing poker in the Nuttall & Mann's saloon in Deadwood, Dakota Territory by a man named Jack McCall. Some claim McCall's motive was because Hickok may have killed McCall's brother, but that's never been confirmed.
Anyway, the only firearm that was found on Hickok's body was a S&W No 2 Old Army. It was confiscated by the sheriff along with Hickok's other effects. At some point a local shopkeeper complained to the sheriff that Hickock had run up a tab at his store and the revolver was given to the shopkeeper.
It was handed down through the family of that store owner and with documentation finally came up at auction in 2013.
The bidding reached $220,000 but the family had a higher reserve and it did not sell. There's been no news on it since.
Lastly in 1868 about 1500 Model 2's were purchased and shipped to Japan to be used by officers in the Boshin war.
This was depicted in the Fall of the Samurai expansion to Shogun 2 Total war.
Each officer of the modern infantry units are depicted as using a model 2
and here's a young Samurai with a holstered model 2
Your turn Sar1n