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Categories are defined for the Vat (Speed at threshold) which is Vso multiplied by 1.3, or Vsldg multiplied by 1.23 .
Since these speeds are related to the weight of the aircraft is it possible for an aircraft to start the flight in one category but then move into another one?

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The FAA's Instrument Procedures Handbook, Chapter 4 has a good explanation of this (emphasis mine):

An airplane is certified in only one approach category, and although a faster approach may require higher category minimums to be used, an airplane cannot be flown to the minimums of a slower approach category. The certified approach category is permanent and independent of the changing conditions of day-to-day operations.

And:

A pilot must use the minima corresponding to the category determined during certification or higher. Helicopters may use Category A minima. If it is necessary to operate at a speed in excess of the upper limit of the speed range for an aircraft’s category, the minimums for the higher category must be used. For example, an airplane that fits into Category B, but is circling to land at a speed of 145 knots, must use the approach Category D minimums. As an additional example, a Category A aircraft that is operating at 130 knots on a straight-in approach must use the approach Category C minimums.

In other words, the aircraft's "certified approach category" never changes. However, for operational reasons a pilot may use a higher category's minimums for an approach, but never a lower category's. One possible scenario would be if the flaps are inoperative and the pilot can't slow the aircraft to its normal approach speed.

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Here is the wording (emphasis mine) adopted by ICAO (Doc 8168, Vol1, Section4, 1.3 - Categories of aircraft):

The criterion taken into consideration for the classification of aeroplanes by categories is the indicated airspeed at threshold (Vat), which is equal to the stall speed Vso multiplied by 1.3, or stall speed Vs1g multiplied by 1.23 in the landing configuration at the maximum certificated landing mass. If both Vso and Vs1g are available, the higher resulting Vat shall be applied.

So the category of a given aircraft doesn't depend on its current weight but on its maximum certificated landing mass, which is constant.

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