I came across the following grammatical terms and example sentences on Wikipedia:
As an adjectival phrase modifying a noun phrase that is the object of a verb, provided the verb admits this particular construction:
I saw them digging a hole. We prefer it standing over there.
As an adverbial phrase, where the role of subject of the nonfinite verb is usually understood to be played by the subject of the main clause. A participial clause like this may be introduced by a conjunction such as when or while:
Looking out of the window, Mary saw a car go by. We peeled the apples while waiting for the water to boil.
Why are the first two sentences said to have adjectival phrases? I think they are adverbial phrases because they are related to time, not quality. And how are the first two sentences different from the last two?