You can't just pump the cold water through the heating elements, because of condensation. The correct installation, requires special heating, cooling elements that include fans (it helps with the cooling) and drainage for the condensation.

I have "cooled floors", so basically heated floors but you can reverse the circuit in summer. Condensation really isn't an issue unless the temperature difference is massive, say room temperature being 28 degrees C and then kicking the system into high gear trying to get to 20 degrees C.

Once room and floor temperature match, condensation doesn't occur anymore as long as you keep it running.