Note: Loss of the system A engine-driven hydraulic pump, and a heavy demand on system A, may cause A/P A to disengage.
Just finding it difficult to understand why the A/P would disengage in this case.
The Boeing 737 has two independent autopilots that use two independent hydraulic servos for controlling the aircraft:
Autopilot Flight Director System (AFDS)
The AFDS is a dual system consisting of two individual flight control computers (FCCs) and a single mode control panel. The two FCCs are identified as A and B. For A/P operation, they send control commands to their respective pitch and roll hydraulic servos, which operate the flight controls through two separate hydraulic systems.
(Boeing 737 NG FCOMv2 4.20.1 - Automatic Flight - System Description, emphasis mine)
You can see the two autopilots A and B connected to hydraulic systems A and B in the hydraulic system overview:
(Boeing 737 NG FCOMv2 13.20.1 - Hydraulics - System Description)
Without the engine-driven hydraulic pump, the respective hydraulic system relies on the AC electrical pump to pressurize the system:
A and B Hydraulic System Pumps
Both A and B hydraulic systems have an engine–driven pump and an AC electric motor–driven pump. [...] An engine–driven hydraulic pump supplies approximately 4 times the fluid volume of the related electric motor–driven hydraulic pump.
(Boeing 737 NG FCOMv2 13.20.2 - Hydraulics - System Description)
Since the engine-driven pump can supply 4 times the fluid volume of the electric pump, the system pressure can drop during times of heavy demand (e.g. gear operation or flap extension/retraction) when only the electric pump is available. The respective autopilot servos can therefore (temporarily) not function as intended and the autopilot will disengage itself.
This could in principle affect both system A and B. However, the PTU (power transfer unit) can use system A hydraulic pressure to assist system B:
Power Transfer Unit
The purpose of the PTU is to supply the additional volume of hydraulic fluid needed to operate the autoslats and leading edge flaps and slats at the normal rate when system B engine–driven hydraulic pump volume is lost. The PTU uses system A pressure to power a hydraulic motor–driven pump, which pressurizes system B hydraulic fluid. The PTU operates automatically when all of the following conditions exist:
- system B engine–driven pump hydraulic pressure drops below limits
- flaps are less than 15 but not up.
(Boeing 737 NG FCOMv2 13.20.4 - Hydraulics - System Description)
Since these conditions are met during flap and slat operation (time of heavy demand on system B), the PTU would likely provide enough pressure to keep autopilot B operational after engine-driven pump 2 has failed. Unlike the PTU on the A320, the system does not work in the other direction. Low pressure in system A is not increased by the PTU and therefore autopilot A is at a higher risk of failing during electric hydraulic pump operation.