Learn Hardcore Hungarian: A könyv az övé, nem az enyém.

a before consonant-initial words: a könyv
az before vowel-initial words: az övé, az enyém In this sentence, the first article belongs to könyv, and the second one is required before the standalone possessive forms övé/enyém.

Is the second az a demonstrative (“that”), or just the article?

Here it’s the definite article. If you wanted “That is his/hers,” you would say: Az az övé (first az = that, second az = the article before övé).

Why is there no verb like “van” (to be)?

In present tense, 3rd person, Hungarian drops the copula in equational sentences. So you say A könyv az övé, not A könyv az övé van. You use van with locations, adverbs, times, etc.:

A könyv otthon van. (The book is at home.)
A könyv nem otthon van. (The book is not at home.)

What exactly is övé, and why not őé?

övé is the standalone possessive meaning “his/hers (one).” It’s formed with a linking -v in the 3rd person: stem öv- + -é → övé. The form őé is incorrect.
The set of standalone possessives is:

enyém (mine)
tiéd (yours, sg. informal)
övé (his/hers)
miénk (ours)
tiétek (yours, pl.)
övék (theirs)

Why is there az before enyém in the second clause too?

Standalone possessive forms normally take the definite article: az enyém, a tiéd, az övé, etc. You generally cannot drop it: say nem az enyém, not “nem enyém.”

Could/should I use hanem (“but rather”) here?

You use hanem after a negated clause to offer the correct alternative:

A könyv nem az enyém, hanem az övé.
Your original sentence is also fine: A könyv az övé, nem az enyém. It presents an assertion plus a contrasting denial.

Can I reverse the order to Az övé a könyv? Does it change the meaning?

Yes: Az övé a könyv is also correct. Word order sets topic/emphasis.

A könyv az övé: topic = the book; comment = it belongs to him/her.
Az övé a könyv: topic = whose it is; comment = the book is the one that’s his/hers.
Both mean the same fact; the nuance depends on what’s given/new in context.

Is övé gendered? How do I make it clear it’s “his” vs “hers”?

Hungarian doesn’t mark gender here; övé can mean “his” or “hers.” If you need to disambiguate, add a noun or name:

a férfié / a fiúé (the man’s / the boy’s)
a nőé / a lányé (the woman’s / the girl’s)
Péteré (Peter’s)

How do I say “The books are his/hers”?

Use the plural form of the standalone possessive:

A könyvek az övéi. (books = plural; övéi = “his/her ones”)
Similarly: A tollak az enyéim. (The pens are mine.)

How do I ask “Whose is the book?” and answer?

Question: Kié a könyv?
Answer: Az övé. / Nem az enyém.
Use kié to ask about possession with these -é forms.

Can I put case endings on these forms (with mine, from his/hers, etc.)?

Yes, they inflect like nouns:

az enyémmel (with mine)
az övéből (from his/hers)
az övében (in his/hers)
az enyémért (for mine)

Can I drop the article and say A könyv övé?

No. The standalone possessive needs the article: A könyv az övé. Likewise in the negative: A könyv nem az enyém.

Why does övé have ö, not long ő like the pronoun ő?

In 3rd person possessive stems, Hungarian inserts a linking -v and the long ő becomes short ö: ő → öv-. Then add -é → övé; plural possessee → övéi.

How would I say “The book is Peter’s, not mine”?

A könyv Péteré, nem az enyém.
The suffix -é attaches to names and nouns to mean “X’s (one).”

Pronunciation tips for the sentence?

ö: front rounded vowel (lip-rounded “e”), as in könyv, övé.
é: long close-mid e, as in övé, enyém.
ny: a single palatal sound [ɲ], like Spanish ñ in señor.
könyv: syllables: kö-nyv; final v is pronounced; ny is one sound.

How do I say “The book is yours” formally?

Use the formal pronoun Ön: A könyv az Öné.
And the contrast: A könyv az Öné, nem az enyém. (Capitalize Ön/Öné in formal writing.)