In your example, fat is an appositive adjective. In general, these are appositive phrases.

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Appositives
An appositive is a word or group of words that describes or renames a noun or pronoun. Incorporating appositives into your writing is a useful way of combining sentences that are too short and choppy. An appositive may be placed anywhere in a sentence, but it must come directly before or after the noun to which it refers:

Appositive after noun: Scott, a poorly trained athlete, was not expected to win the race.

Appositive before noun: A poorly trained athlete, Scott was not expected to win the race. Unlike relative clauses, appositives are always punctuated by a comma or a set commas. Take a look at the way Naomi uses appositives to include additional facts in her essay.


https://pressbooks.pub/theworryfreewriter/chapter/mini-gramm...

An appositive is a noun or pronoun — often with modifiers — set beside another noun or pronoun to explain or identify it. Here are some examples of appositives (the noun or pronoun will be in blue, the appositive will be in red).

Your friend Bill is in trouble.
My brother's car, a sporty red convertible with bucket seats, is the envy of my friends.
The chief surgeon, an expert in organ-transplant procedures, took her nephew on a hospital tour.

https://owl.purdue.edu/owl/general_writing/grammar/appositiv...

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An Appositive Adjective is a traditional grammatical term for an adjective (or a series of adjectives) that follows a noun and, like a nonrestrictive appositive, is set off by commas or dashes.

Appositive adjectives often appear in pairs or groups of three (tricolons).

Examples and Observations
"Arthur was a big boy, tall, strong, and broad-shouldered."
(Janet B. Pascal, Arthur Conan Doyle: Beyond Baker Street. Oxford University Press, 2000)
"No Chinese emperor was more resplendently arrayed. As for the cigarette that he holds out, half smoked, to be taken and deposited by his valet, a whole civilization—urbane, authoritative, preposterous, and doomed—resides in that single gesture."
(Anthony Lane, "Life and Death Matters." The New Yorker, February 8, 2010)
"Much of the greatest poetry, ancient and modern, has been occupied with a similar image: the figure of the abandoned woman."
(Lawrence Lipking, Abandoned Women and Poetic Tradition. The University of Chicago Press, 1988)

https://www.thoughtco.com/what-is-appositive-adjective-16889...

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What Is an Appositive Phrase?
An appositive phrase redefines, renames, or modifies a noun with essential or nonessential information. Basically, appositives clarify a noun with another noun or noun phrase that gives a noun extra context. Appositives can be a single word or a group of words and are often enclosed in commas if they come in the middle of a sentence.

https://www.masterclass.com/articles/appositive-phrase-guide