A prepositional phrase is a phrase that contains the following types of words:
A preposition, that are function words used to link nouns, pronouns, or phrases to other words within a sentence. Examples of prepositions are <em>without, in, up, after, before, at, with, for, through.</em>
The object of the preposition, like nouns, pronouns, gerunds, etc.
And one or more than one modifiers of the object (Sometimes, not always), such as "a" and "the".
So, "without a coat" is a prepositional phrase because it has all those elements: It has the preposition "without" that indicates the absence or lack of something ("a coat"), it has the object of that preposition which it's "coat", and it also contains a modifier of that object which it's "a"
