Hyundai Repair: 2003 Tiburon 6-speed hard to shift, hyundai tiburon, 2003 hyundai tiburon

Question
QUESTION: I have a 2003 Hyundai Tiburon with the 6-speed manual transmission. It has started to be difficult to shift into gear. The dealer replaced the master cylinder but the problem still exists. There is no noticeable noise. It happens when the car has been sitting for a while and the engine is cold. Once the engine warms up, it is back to normal. The dealer is still scratching his head after 2 days of diagnostics. They are hinting at a clutch replacement. With the clutch pedal in and the car in gear at a light, there is no tugging or noticeable engine drag. Any clues you might be able to provide would be greatly appreciated.
ANSWER: It's possible the wrong weight oil was put in the transmission.  Check to make sure the oil used was 70w-90 or 75w-90.  If in doubt, change it.

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QUESTION: Could the oil have degraded over time? The car has 53000 miles on it and the transmission oil hasn't been changed.  

Answer
It shouldn't have degraded.  My primary reason for suspecting the oil was that you indicated the problem occurred when the engine was cold.  Too thick an oil will make the transmission difficult to shift when cold.

Based on your description, I don't particularly think this is a clutch or synchronizer issue.  These would tend to cause grinding when you actually get the gears to shift.  I'm presuming the difficulty you're having is that the lever is difficult to move.  

This is one of those situations where someone knowledgeable and experienced checking the vehicle is in a better position to judge than I.  I cannot feel the problem or check the vehicle, whereas the technician checking the vehicle can do both.  I would think, however, that it would be reasonably simple to determine whether the issue was with the clutch or the transmission.  A clutch issue will be nonexistent when the engine is not running.