I am editing an object, use local transform orientation.
Here it is
Why is not Z, the blue arrow pointing "screen upwards"?
That would make life so my easier!
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World Transform and Local Transform exists for reasons.
You can reset/apply your object's transform to make your Local Z axis pointing upward. But there is no way to lock Z axis to point upward. What you can try is to move/rotate your object's mesh in Edit mode, in there, your rotation/move operations won't change the local axis's orientation.