Those areas are never factory finished. They are the internal box color finish, which is usually a natural maple. From the factory, it’s an upgrade to finish them to a matching finish. It would be standard to just finish the same factory finished portions and to not finish those areas. Just like the factory cabinets are currently finished. And, just like the original purchase, finishing those areas would be possible as an upgrade for additional cost. The biggest obstacle to coating them is that they usually are melamine coated, which takes specialty treatment and extra work to allow them to be coated.

This doesn't take any research. Just tell the painter what you want. Simple. Done.

GreenDesigns, sorry my question wasn't clear. These are custom built solid wood cabinets that were stained in a dark stain including the underneath of the top cabinets. What I would like to know is if a professional painter would know to paint the underneath so that it all looks same and makes the cabinet look finished and new.

Only if you specify that in the scope of work, so that the upgrade could be charged. Figure 7-9K on average for a not giant kitchen, for someone doing the right prep, and using a conversion varnish. Larger is more. More than standard work is more.

When bidding the job, this is something the pro should know needs to be painted. If they were not finished by the factory, then I can understand not finishing them, but if they are stained, then they need to be finished. You should still ask them to include it in the price though if you are concerned about it.