To show that an object S has property P, first show that the collection of all objects satisfying P is closed under a bunch of operations, prove that certain very simple objects have property P, and show that S can be "decomposed" or "filtered" or somehow unscrewed into these simple objects using the operations mentioned above.
This is a sort of induction, and it is used all the time to turn annoying verifications into verifying that something is true for like... a point. Maybe a shorthand for this "slick method" would be "think like Grothendieck."
For lots of examples of this see any proof in Higher Topos Theory or Higher Algebra by Lurie.
