Kit Carson comments to Fremont about playing "Taps" the bugle call. "Taps" was first written in 1862 by General Dan Butterfield and soon accepted for lights out and then at funerals. There was a "Tatoo" call used for lights out in 1850 and the army also used a French set of bugle calls, but not yet had "Taps" at the time of the setting of the movie.
At the very beginning a group of men see the distinct upside "U" shape of horse shoes. One announces that they are signs of Indians. Native-Americans at that time did not shoe their horses.
In real life, Kit Carson never learned to read or write. In two scenes, he's shown reading something, and also wrote Dolores a letter.