Canada's Westjet operates the 737-200 & 700. The only time the PF role is mandated to be handled by the Captain is for the landing from a Pilot Monitored Approach (mandatory if the RVR is less than 2600 ft). The FO is the Pilot Flying for the approach down to minimums, selecting and monitoring the autopilot.
There is no differentiation between Captain and First Officer using the autopilot.