If you have to chose some auxiliary object and that object is not unique, it's better to make all choices simultaneously.
I think there are many examples of this, but for me it first hit home when I learned about crystalline cohomology. There you want to lift varieties in positive characteistic to characteristic zero. Locally there are many nonisomorphic lifts, and rather than picking one, it's better to work with the category of all of them. I've absorbed this lesson pretty fully, to the point where I don't need to remind myself of it, but at first it seemed revolutionary.