In 2001, GM  changed the transmission cooler mid model year. At this time, the connection fitting switched from the old flare tube and nut to a newer metric quick connect fitting. One way to identify which model you have is look at the retaining nut holding the cooler in the tank. The first design flare fitting cooler is held in with a thin wall stamped steel nut and the second design is a solid aluminum nut.

The new fitting is hard to figure out because the retaining clip is hidden inside. A special tool is required to remove the fitting from the tube. This was somewhat a transitional fitting as GM moved toward quick connect fittings. The later C6 fitting has the clip exposed and it is much easier to access. 

Since there are two possible radiators for the 1997-2004 Corvette with automatic transmission it is possible to purchase the wrong replacement and the transmission lines will not match. If your Corvette has the 1st design (Flare fitting) and you purchased a 2nd design (metric) replacement radiator DeWitts offers an adapter part number 876F. 

Unfortunately there is no direct conversion fitting for the 2nd design (metric) to a 1st design (flare) radiator. In this case you will either have to return the radiator and purchase the correct one or cut the steel tube and flare the line. See Earls fitting 961947.

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