What is the grammatical role of "for you" in "I am waiting for you"? Is it a direct object, a prepositional phrase or what? Functionally, it seems to be of an object, as "you" in "I love you", but I am not sure how English labels objects of verbs which require a preposition.
The grammatical role of for you in
- I am waiting for you.
It’s also an argument here:
- She came looking for you.
- I made a cake for you.
If in #2 for you seems more obviously an argument than it does in #3, remember that #3 is just the dative alternation for the ditransitive two-object version:
- I made you a cake.