You are correct. (see below copy of 121 appx. H)
I'm not sure about the 2500 hour requirement for a Part 142 school though. I'll have to look that up.
I did not remember ever seeing that requirement for 121.
JAFI
Part 121 Appx. H
Level C
Training and Checking Permitted
1. For all pilots, transition training between airplanes in the same group, and for a pilot in command the certification check required by §61.153(g) this chapter.
2. Upgrade to pilot-in-command training and the certification check when the pilot—
a. Has previously qualified as second in command in the equipment to which the pilot is upgrading;
b. Has at least 500 hours of actual flight time while serving as second in command in an airplane of the same group; and
c. Is currently serving as second in command in an airplane in this same group.
3. Initial pilot-in-command training and the certification check when the pilot—
a. Is currently serving as second in command in an airplane of the same group;
b. Has a minimum of 2,500 flight hours as second in command in an airplane of the same group; and
c. Has served as second in command on at least two airplanes of the same group.
4. For all second-in command pilot applicants who meet the aeronautical experience requirements of §61.159 of this chapter in the airplane, the initial and upgrade training and checking required by this part, and the certification check requirements of §61.153 of this chapter.