Here are the characteristics of the devices that control starting and warm-up on K-jet, in the Reader's Digest Condensed Version.
Initial starting (first running of the engine) is controlled by the start valve, and if it fails (thermo-time switch usually) then the engine doesn't start from dead cold, or only after prolonged cranking.
If the engine runs poorly AFTER starting, and has a weak, low speed idle, then the control pressure regulator is not enrichening the mixture sufficiently. Test for this by rapidly opening the throttle. If the engine spits back, or stalls, its lean.
If the engine runs slowly AFTER starting, but has a strong low speed idle, then the auxiliary air valve has failed. Test for this by rapidly opening the throttle. If the engine accelerates well, then them mixture is correct, but the total air flow is low.
Sometimes a poor idle during warm-up is due to worn injectors, as when the engine is cold, mixture formation is more important than when hot. Testing the pattern is easy, but not part of the condensed version.
Adjusting the control pressure regulator is possible, but not for the inexperienced.
Don't buy any parts unless the mechanic in question seems fully confident. Hanging parts isn't diagnosis, or a replacement for it.