More or less. A different part of the weapon morphs each time you do an upgrade after you open the melee and ranged chests on the Road to Rule. That which affects the morph is chosen from a pool based on what you have done up to that point. Some think the pool is completely random, but my experience suggests that there is some priority system with a bit of randomization-- like a percentage chance.
However the system works, four morphs will be chosen from the various pools based on your actions. A fully upgraded weapon will have a list of 4 things you did and how they affected your weapon.