Usually, the filter is before the cooler to prevent trash from accumulating in the core. An "H" bypass line between them and returning into the cooler's return allows the cooler to be shut down at low temperatures.
The cooler must have an air pressure differential between the front and back, even with a fan. A fan-equipped cooler hung in the engine bay will simply recirculate air around the cooler and back again (like a fan with no shroud).
