Después de haber mirado el mapa, encontramos la farmacia sin problema.

Both are possible.

• después de mirar el mapa = after looking at the map
• después de haber mirado el mapa = after having looked at the map

The version with haber + past participle makes it extra clear that the action of looking at the map was completed before the next action happened. In practice, both are natural, but haber mirado sounds a bit more explicit about the sequence.

It is the perfect infinitive.

It is formed like this:

• haber
  • past participle
• haber mirado = to have looked / having looked

So in después de haber mirado, the phrase means after having looked. Even though mirado is a participle, the whole structure acts like a non-finite verb phrase.

Because after the preposition de, Spanish normally uses an infinitive, not a gerund.

So:

• correct: después de haber mirado
• not correct here: después de habiendo mirado

Habiendo mirado can exist in Spanish, but not after después de in this structure.

Because mirar and ver are related but not identical.

• mirar = to look at, to examine, to check
• ver = to see

With a map, Spanish usually uses mirar because you actively look at it to get information. So mirar el mapa feels very natural, like look at/check the map.

Here it is preterite, so it means we found.

The tricky part is that encontramos has the same form in:

• present: we find
• preterite: we found

Context tells you which one it is. Since the sentence begins with Después de haber mirado el mapa..., it clearly describes past events, so encontramos is understood as we found.

Because Spanish often leaves out subject pronouns when they are not needed.

The ending -amos in encontramos already tells you the subject is we. So:

• (Nosotros) encontramos la farmacia...

Both are possible, but Spanish usually prefers the version without nosotros unless you want emphasis or contrast.

La farmacia suggests a specific pharmacy:

• the one they were trying to find
• the one they needed
• the one already understood from the situation

If you said una farmacia, it would sound more like a pharmacy / any pharmacy. The definite article here makes it feel like a known destination rather than just any pharmacy.

Sin problema means without any problem or more naturally easily / with no trouble.

It is a common idiomatic expression. Spanish often uses this kind of structure without an article:

• sin problema
• sin dificultad
• sin duda

You may also hear sin problemas, which is also common. In this sentence, sin problema simply means they found it easily.

The comma separates the introductory time phrase from the main clause:

• Después de haber mirado el mapa,
• encontramos la farmacia sin problema.

This is very common and helps readability. In short sentences, punctuation choices can vary a bit, but the comma here is perfectly natural.

Yes. That is a very natural translation.

Even though mirar literally means to look at, with el mapa it often has the sense of check/consult the map. And sin problema can be translated in several natural ways:

• without any problem
• with no trouble
• easily

So After checking the map, we found the pharmacy easily is an excellent English rendering.