Hello everybody,
This question really attracted my attention and I can't help posting some observations,since I've been reading about this point lately.I hope I'll be forgiven for venturing to discuss topic in spite of being still a learner.(Also I hope that,this being my first time to post "explanation",it will be considered as an extenuating circumstance

).And,of course,I'M NOT A TEACHER,I'M NOT EVEN AN ADVANCED LEARNER
There is a great deal of different terms for the element which follows a preposition in the prepositional phrase,but through reading of Quirk/Greenbaum grammar(mostly) and with a great help of Soup,here on the forum,I think I've managed to get their point on this.This two authors use the term "prepositional object" for the element of the sentence structure,putting it in the same level with the direct and indirect object.They indicate the term "oblique object" as an alternative term.When analysing structure of a prepositional phrase they use the term "prepositional complement"for the element that follows a preposition in the prepositional phrase (saving the term "object" exclusively for the element of the sentence structure,and with that terminology I suppose they keep a distinction between phrase constituents and sentence constituents in the analysis of the sentence structure.So,a noun or noun phrase which follows a preposition in the prepositional phrase is called a prepositional complement which may be part of an adverbial ,as in the sentence which Cooklava has posted,or it may be a prepositional object as in the following sentences taken again from Quirk/Greenbaum grammar :
1. I applied for a grant ("applied for"- monotransitive prepositional verb, "a grant" - prepositional object)
2. Nobody will blame you for the mistake ("you" - direct object,"blame"- doubly transitive prepositional verb (i.e takes two objects) ,"the mistake"- prepositional object introduced by the preposition "for")
As far as I could understand,for the correct structural analysis of this part of the sentence,it is a crucial point to grasp the notion of "prepositional verb" as a more or less idiomatic part (not a free combination of a verb and preposition) of the sentence ,analogue to "phrasal verb".I've found also that,in the analysis of the sentence(sentences similar to those quoted above),the whole prepositional phrase is analysed by some grammarians as a separate part and is termed just like that,or alternatively as a "prepositional complement",and the verb is analysed separately,as a lexical verb,without a preposition.However,I find(after a good deal of hard thinking

) Quirks explanation quite understandable and logical.
I want to say in addition that I think that a good grasp of some basic terms in grammar is very convinient if not necessary element in acquiring more vocabulary and the language as a whole.
Best regards
Velimir