A kölcsönzés gyors.

Because Hungarian usually drops the verb to be in the present tense, third person when the predicate is a noun or an adjective.

So:

• A kölcsönzés gyors. = The borrowing is fast.

Hungarian does not normally say A kölcsönzés van gyors.

But in other tenses, to be does appear:

• A kölcsönzés gyors volt. = The borrowing was fast.

And with other persons, forms of to be are also used:

• Gyors vagyok. = I am fast.
• Gyors vagy. = You are fast.

So this sentence is completely normal Hungarian.

A is the definite article, so it means the.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since kölcsönzés begins with k, which is a consonant, you use a:

• a kölcsönzés

If the next word started with a vowel, you would use az instead:

• az autó = the car

Kölcsönzés is a noun meaning something like:

• borrowing
• lending
• loaning
• sometimes rental / renting

The exact English translation depends on the context.

That is because the Hungarian word family around kölcsön often covers ideas that English separates more clearly into borrow, lend, and sometimes rent.

So in a sentence like A kölcsönzés gyors, the meaning is often something like:

• The borrowing process is fast
• The loan service is fast
• The rental process is fast

The context tells you which English wording fits best.

It comes from the verb kölcsönöz, which means to lend / to borrow / to loan, depending on context.

Then Hungarian adds the noun-forming ending -és:

• kölcsönöz → kölcsönzés

This -és ending often creates a noun meaning the act or process of doing something.

A rough English comparison would be:

• to borrow / lend → borrowing / lending

So kölcsönzés is basically the act/process of borrowing or lending.

Because here gyors is not directly describing the noun inside a noun phrase. It is the predicate adjective of the whole sentence.

Compare these two patterns:

• a gyors kölcsönzés = the fast borrowing
  Here gyors comes before the noun because it directly modifies it.

• A kölcsönzés gyors. = The borrowing is fast.
  Here gyors comes after because it is the statement being made about kölcsönzés.

So:

• gyors kölcsönzés = fast borrowing
• A kölcsönzés gyors = The borrowing is fast

That difference is very important in Hungarian.

Because gyors is an adjective, and in this sentence it describes the noun kölcsönzés.

• gyors = fast
• gyorsan = quickly / fast as an adverb

Here the sentence means:

• The borrowing is fast

So you need the adjective gyors.

You would use gyorsan if you were describing how an action happens:

• Gyorsan megy a kölcsönzés. = The borrowing process goes quickly.

So:

• A kölcsönzés gyors. = adjective
• A kölcsönzés gyorsan megy. = adverb

Yes. Hungarian word order is flexible, and changing it often changes the emphasis.

• A kölcsönzés gyors.
  This is a neutral statement: The borrowing is fast.

• Gyors a kölcsönzés.
  This puts more focus on fast. It can sound like: The borrowing is fast or The borrowing is fast, actually.

In Hungarian, word order is often about information structure, not just grammar. So both are grammatical, but they may feel slightly different in emphasis.

No. Here kölcsönzés is in the basic dictionary form, the nominative, because it is the subject of the sentence.

So:

• A kölcsönzés gyors. = The borrowing is fast.

There is no special case ending because nothing is happening to the noun in this sentence. It is simply the thing being talked about.

If it had another role, it could take a case ending, for example:

• a kölcsönzésről = about the borrowing
• a kölcsönzéshez = for / to the borrowing
• a kölcsönzést = the borrowing as a direct object

A simple learner-friendly pronunciation guide would be:

• a KÖL-chön-zaysh gyors

A few useful sound notes:

• ö is similar to German ö or French eu
• cs sounds like ch in chair
• zs sounds like the s in measure
• gy is a soft consonant that does not exist exactly in English; learners often approximate it at first
• Hungarian stress is usually on the first syllable of each word:
  • A KÖLcsönzés GYORS

So the hardest word here is usually kölcsönzés.