Die Bibliothek bestätigt die Vormerkung.

Because both Bibliothek and Vormerkung are feminine singular nouns.

In this sentence:

• die Bibliothek = the library
• die Vormerkung = the reservation / hold request

For feminine singular nouns, die is used in both the nominative and accusative cases, so the article looks the same for both the subject and the direct object.

That is why you see die twice.

In Die Bibliothek bestätigt die Vormerkung:

• die Bibliothek is the subject
• die Vormerkung is the direct object

A big clue is word order:

• In a normal German main clause, the subject often comes first.
• The finite verb comes second.
• The object often comes after the verb.

So the structure is:

• Die Bibliothek = subject
• bestätigt = verb
• die Vormerkung = object

Also, semantically it makes sense: the library is doing the confirming.

bestätigt is the 3rd person singular present tense form of bestätigen.

So:

• ich bestätige = I confirm
• du bestätigst = you confirm
• er/sie/es bestätigt = he/she/it confirms

Here, die Bibliothek is grammatically singular, so the verb must also be singular:

• Die Bibliothek bestätigt = The library confirms

Because with feminine singular nouns, the article die is the same in both the nominative and accusative.

Compare:

• nominative feminine: die
• accusative feminine: die

So:

• die Bibliothek could be nominative
• die Vormerkung could be accusative

You cannot tell the case from die alone in this sentence. You mostly rely on:

1. word order
2. meaning
3. the verb’s pattern

This is very common in German.

In a library context, Vormerkung usually means something like:

• reservation
• hold
• advance request for an item

So if a book is unavailable, a user can place a Vormerkung, and later the library confirms it.

A native English speaker may find reservation or hold request more natural depending on the library system.

Because German main clauses follow the verb-second rule.

In a normal statement, the finite verb goes in position 2:

• Die Bibliothek | bestätigt | die Vormerkung

Even if some other element comes first, the verb still stays second:

• Heute bestätigt die Bibliothek die Vormerkung.
• Per E-Mail bestätigt die Bibliothek die Vormerkung.

So in your sentence, bestätigt is second because that is standard German word order.

In German, all nouns are capitalized, not just proper names.

So:

• Bibliothek is capitalized because it is a noun
• Vormerkung is capitalized because it is a noun

This is one of the most noticeable differences from English spelling.

Usually, yes. Bibliothek normally means library.

Just be careful not to confuse it with English bibliography or with German Buchhandlung (bookstore).

So here:

• Die Bibliothek bestätigt ... = The library confirms ...

That is the natural meaning.

German present tense often covers both:

• simple present: confirms
• present progressive: is confirming

So Die Bibliothek bestätigt die Vormerkung can mean:

• The library confirms the reservation
• The library is confirming the reservation

Which one sounds best in English depends on context, but the German form itself is just the normal present tense.

A simple approximate pronunciation is:

dee bee-blee-oh-TEHK beh-SHTEH-tigt dee FOR-mer-koong

A few helpful notes:

• Bibliothek is stressed on the last syllable: biblioTHEK
• bestätigt has the stressed stä
• Vormerkung is stressed on Vor
• German r may sound different from English r
• the final -ung in Vormerkung sounds like -oong, not like English -ung in sung

If you want a more exact pronunciation, the IPA is:

[diː biblioˈteːk bəˈʃtɛːtɪçt diː ˈfoːɐ̯ˌmɛʁkʊŋ]

Yes, but the verb still has to stay in second position in a main clause.

For example:

• Die Bibliothek bestätigt die Vormerkung.
• Die Vormerkung bestätigt die Bibliothek. — this changes the meaning and sounds odd
• Heute bestätigt die Bibliothek die Vormerkung.
• Per E-Mail bestätigt die Bibliothek die Vormerkung.

So while German allows some flexibility, the original order is the most natural and clear for this sentence.

The dictionary forms are:

• die Bibliothek = library
• bestätigen = to confirm
• die Vormerkung = reservation / hold request

When learning German, it is useful to memorize nouns with their article:

• die Bibliothek
• die Vormerkung

That helps you remember grammatical gender and use the correct articles later.