Everyone expected the chestnut to win.
I think it's a noun but it's pronoun someone said..
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It's a noun (or a noun phrase formed from a determiner and a noun)
It's not a pronoun, as it is not one of the words "he, she, it, theirs..." that refer anaphorically (or otherwise) to another noun. Instead it names or refers to a horse.
It's not an adjective, although it has been formed from an adjective. It is often possible to take an adjective like "chestnut" and use it as a noun "the chestnut" to mean "the chestnut thing" (and in this context I suppose the "thing" is "horse")
The pronouns in English form a small and closed class of words. "Chestnut" is not in that class. One can't use other words as pronouns in the way that one can use nouns as verbs, or adjectives as nouns (for example) So if a word isn't on the list of pronouns, then it isn't a pronoun.