Az esernyő az ajtó mellett van, a sapka pedig a táskában.

Hungarian has two forms of the definite article:

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

So:

• esernyő begins with a vowel sound, so it becomes az esernyő
• ajtó also begins with a vowel sound, so it becomes az ajtó
• sapka begins with a consonant sound, so it becomes a sapka
• táska begins with a consonant sound, so it becomes a táska, and in the sentence that appears as a táskában

This is similar to English a/an, but Hungarian uses the distinction only for the definite article.

That is a very common and important question.

Hungarian often omits van/vannak in the present tense when one noun or adjective simply identifies or describes something:

• A sapka piros. = The hat is red.
• Ő tanár. = He/She is a teacher.

But when you say where something is, Hungarian normally does use van:

• Az esernyő az ajtó mellett van.
• A sapka a táskában van.

So in this sentence, van appears because the sentence is about location.

Mellett means next to / beside.

Hungarian often uses postpositions instead of prepositions.
A postposition comes after the noun, not before it.

So:

• English: next to the door
• Hungarian: az ajtó mellett = literally the door next-to

That is why mellett follows ajtó.

Because Hungarian expresses location in two different ways here:

1. With a case ending
2. With a postposition

1. a táskában

Here -ban is a case ending meaning in.

• táska = bag
• táskában = in the bag

Since -ban is attached directly to the noun, it becomes one word.

2. az ajtó mellett

Here mellett is a separate postposition meaning beside / next to.

• ajtó = door
• ajtó mellett = beside the door

So one is written as a single word because of a suffix, and the other is written as two words because of a separate postposition.

-ban / -ben is the inessive case, which usually means in or inside.

Examples:

• házban = in the house
• dobozban = in the box
• táskában = in the bag

The form depends on vowel harmony:

• -ban after back vowels
• -ben after front vowels

Because táska has back vowels, it takes -ban:

• táska → táskában

So a táskában means in the bag.

Pedig is a very useful word. Here it adds a sense like:

• while
• whereas
• as for
• and ... meanwhile / on the other hand

In this sentence:

• Az esernyő az ajtó mellett van, a sapka pedig a táskában.

it helps contrast the two items:

• the umbrella is in one place
• the hat, on the other hand, is in another place

So pedig does not radically change the meaning, but it makes the second part feel more contrastive or more clearly paired with the first.

A natural English feeling might be:

• The umbrella is by the door, and the hat is in the bag.
• or The umbrella is by the door, while the hat is in the bag.

Yes, it is omitted there because Hungarian often leaves out repeated material when it is obvious.

The full version would be:

• Az esernyő az ajtó mellett van, a sapka pedig a táskában van.

But since the second van is easy to understand from context, Hungarian can leave it out.

So both of these are possible:

• ... a sapka pedig a táskában.
• ... a sapka pedig a táskában van.

The shorter version is very natural.

The word order can change, but the emphasis changes too.

The given sentence:

• Az esernyő az ajtó mellett van, a sapka pedig a táskában.

is a neutral, natural way to present where each item is.

But Hungarian word order is flexible and often reflects topic and focus.

For example:

• Az ajtó mellett van az esernyő.
  = It is by the door that the umbrella is.
  This puts more attention on the location.

• A táskában van a sapka.
  = The hat is in the bag. with emphasis on in the bag

So yes, the order can change, but not randomly. It changes what feels emphasized.

Because they are the subjects of the sentence, and the basic subject form in Hungarian has no special ending. This is the nominative form.

So:

• esernyő = umbrella
• sapka = hat/cap

In this sentence, they are simply the things being talked about, so they stay in their basic form.

By contrast, the location word can get an ending:

• táska → táskában

or appear with a postposition:

• ajtó mellett

Hungarian normally uses articles quite regularly, and each noun phrase usually has its own article when it is definite.

So the sentence naturally has:

• Az esernyő
• az ajtó mellett
• a sapka
• a táskában

This is normal Hungarian structure. English can sometimes sound fine with fewer articles in shortened expressions, but standard Hungarian usually keeps them.

Sapka often means cap or a soft hat, especially an everyday one. In many beginner translations it may just be given as hat.

So depending on context, a sapka could be:

• the cap
• the hat

If you wanted a broader or more formal word for hat, Hungarian also has kalap, which usually means a brimmed hat.

So:

• sapka = cap / knit hat / casual hat
• kalap = hat, especially with a brim

It is completely natural. It sounds like a normal descriptive sentence telling you where two objects are.

A full version with both verbs would also be natural:

• Az esernyő az ajtó mellett van, a sapka pedig a táskában van.

But the version you have is especially nice because it avoids repeating van unnecessarily.

So for a learner, this is a very good example of:

• definite articles (a / az)
• a postposition (mellett)
• a location ending (-ban)
• optional omission of repeated van
• the contrastive word pedig