Not all Kawi FX engines have an engine oil cooler, it depends on the engine code. The engine code is specified by the mower manufacturer.
If an FX engine has an oil cooler, it is mounted directly above the oil filter on the blower housing where it gets airflow thru it supplied by the rotation of the fan on the flywheel which also forces cooling air by the cylinders.
My Kawasaki FX powered Hustler X-ONE is in storage at the moment so I'm not able to take a pic of it and I'm not positive that it has an engine oil cooler installed on it.
I have pics of the X-ONE some where in my pics, just have to looking for them.