Looks like this is an "oil to air" oil cooler, where the oil is redirected by the adapter through the hose to the radiator.
I have limted prctical knowledge of this kind as I have always used "oil to water" coolers....but here goes.
This cools the oil as it circulates through the oil radiator and then returns back to the engine. Not sure when it is filtered.
With this type of cooler the oil is cooled all the time, as it it continually pumped through the cooler. This can cause problems as it doesn't allow your oil to heat up to operating temperature fast enough, and in some cases it may never heat up. This can affect the viscosity of the oil, as well not boiling off any condensation...kind of like driving your car only on very short trips all the time. I tyhink you should install an oil thermostat between the cooler and engine. The thermostat will on allow oil through the cooler after a specific temp has been reached. There as also coverr plates that stop air through the rad.
Thinking about this, you may not in fact have a problem at all. I say this because you may not be able to feel a noticeable diference between the before and after hoses with your hand alone. I can't tell the difference between say 105 and 115 degrees. I suggest getting a lazer thermometer and checking for a temp difference on the input and out put hoses.
Please feel free to correct me if I am wrong with any of the above those with more experience with these.
Mikey