A start up sheet, AKA "commissioning" sheet is not always provided by the installing contractor. Long story short, it takes all the pertinant readings of the system to document it for future use if there is an ussue.
Something that generally any manufacturer has out there for their contractors to use because when you ask a guy on readings from the system, they generally tell you "good", which could mean 5 different things which don't really mean good. A good contractor IMO will always provide a copy of this sheet for documentation to cover his arse IMO. That said, a lot of HVAC contractors don't use actually use one because you actually have to take the time to fill everything out. That said, on a 3rd coil replacement, they should be using one because sooner or later the homeowner will be screaming for a new unit.
If it was the condensing coil, if you saw them do it, you'd recognize it as it's the coil that "wraps" around the exterior of the unit (which may be guarded by metal as well).
What year was the unit installed?
Model and serial?
What is the contractor charging you on labor to replace the 3rd coil (and what did they charge you on the previous two replacements?)
At this point in time, if installed within 10 years, you need to be documenting everything. I'm guessing the contractor isn't doing all of this for free and sooner or later you're going to want a new unit.
I'd guess it's either a Goodman or JCI package unit?
On a sidenote, a picture is worth a thousand words. Contractor should be taking pics.