the A/T goes into CLB mode automatically normally after pressing e.g. “FLC” after takeoff.
a real life example here: A day in the life of a corporate pilot! Citation Longitude | Vegas - Heber City - YouTube (3:46). Procedure is “flaps up + FLC”
hard to see, but after FLC was pressed, A/T chance to CLB.
So once A/T is in “HOLD” Mode on Takeoff, everythink should work
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