Another stupid question. I always thought these were installed on the suction side (between water pickup and pump) but then I see them advertised with a pressure relief. That would indicate that it would go between the pump and the first cooler?
I've never seen a strainer installed any other place then between the pickup and the pump, would kind of defeat the purpose of the strainer. The reason for the pressure relief valve on the strainer is to regulate the incoming water pressure, using certain water pickups or lower units will have an increased water pressure on the intake side causing hoses to potentially pop off or break water pump housings. When I have installed low water pickups I rig up a homemade gauge inline between the pump and pickup to make sure the pressure isn't too high, if it's high you can trim/adjust the water pickup to lower the pressure . You could also do this with an adjustable pop off valve in the strainer.
A lot of the fast catamarans have this issue and required to bleed off the intake pressure to keep hoses from popping off.
The reason for having a large strainer is to also provide a resivor for the water pumps when you exit the water. Putting them after the pumps would negate this benefit.