That is absolutely true. But if the compression had not increased the temperature of the air, we might not have had to use a second heat exchanger in the first place.
Some of the energy spent by the compressor gets turned into heat and the rest of it is used in decreasing the volume of air. Now this compressed air is suddenly expanded by the turbine which results in a huge loss of temperature and so the air cools down. The heat exchangers are there to assist the turbine by making the air temperature workable. They are not the primary means of cooling.