Nadat het poortje weer open is gegaan, lopen we snel naar de uitgang.

Nadat means after and introduces a subordinate clause.

In Dutch, a subordinate clause usually sends its verbs to the end, so you get:

Nadat het poortje weer open is gegaan

instead of a more English-like order.

A very literal word-for-word version would be something like:

After the gate again open has gone

Of course, that is not natural English, but it helps show the Dutch structure.

Because the sentence begins with the nadat clause, that whole clause takes the first position in the sentence. In Dutch main clauses, the finite verb normally comes in second position.

So after the opening clause, Dutch puts the verb first and the subject after it:

Nadat ..., lopen we snel naar de uitgang.

This is called inversion.
If there were no opening clause, you would say:

We lopen snel naar de uitgang.

Dutch normally uses a comma to separate an introductory subordinate clause from the main clause.

So this is standard:

Nadat het poortje weer open is gegaan, lopen we snel naar de uitgang.

The comma helps show where the nadat clause ends and the main clause begins.

The ending -je is the Dutch diminutive ending. It often means something like small or little.

So:

• de poort = the gate
• het poortje = the little gate / small gate

A very important grammar point: all Dutch diminutives are always het words.

So even if the base noun is a de word, the diminutive becomes het:

• de poort
• het poortje

Not necessarily. In Dutch, diminutives are often used for everyday objects even when the speaker is not really emphasizing small size.

In this kind of sentence, poortje can refer to something like:

• a small gate
• an access gate
• a barrier gate
• a station gate or turnstile

So it is often just the normal natural word for that kind of object.

Here weer means again.

So weer open means open again or open once more.

That tells you the gate had been closed before and then opened another time.

Be careful: weer can also mean weather in other contexts, but not here.

Dutch often expresses this idea with opengaan or open gaan, literally to go open.

So:

• Het poortje gaat open = The gate opens
• Het poortje is open gegaan = The gate opened / has opened

This is an intransitive way to say that something becomes open by itself or changes state.

If you want to say that someone opens it, Dutch more often uses a transitive verb such as openen, opendoen, or openmaken, depending on context.

The perfect tense here uses zijn, not hebben, because gaan commonly takes zijn as its auxiliary.

So in the perfect tense:

• gaan → is gegaan

This is especially common with verbs involving movement or a change of state. Here the gate changes from closed to open, so is gegaan fits naturally.

Because the opening happens before the walking.

With nadat, Dutch often uses the perfect tense to show that the earlier action is already completed before the next action starts.

So the sequence is:

1. the gate opens again
2. then we walk quickly to the exit

That is why open is gegaan works well here.

Naar means to or toward.

So:

• naar de uitgang = to/toward the exit

Uitgang is the noun exit.
So de uitgang means the exit.

Dutch uses naar because the sentence is about movement in the direction of a place.
If you said just uit, that would mean out, not specifically to the exit.

Snel means quickly here, so it modifies lopen.

Dutch adverbs like snel are often placed after the subject/verb part and before a prepositional phrase such as naar de uitgang:

lopen we snel naar de uitgang

That order sounds very natural in Dutch:

• lopen we = verb + subject
• snel = adverb
• naar de uitgang = direction

So the sentence flows in a very typical Dutch way.