Za semaforem je knihovna vlevo.

Because za in this sentence means behind / past / after in a location sense, and with that meaning it takes the instrumental case.

• semafor = traffic light
• za semaforem = behind the traffic light / past the traffic light

So:

• nominative: semafor
• instrumental singular: semaforem

This is a very common pattern in Czech:

• za domem = behind the house
• před školou = in front of the school
• mezi stromy = among the trees

It can be translated in a few ways depending on context. In directions, za semaforem often means something like:

• past the traffic light
• after the traffic light
• literally, behind the traffic light

In English, behind the traffic light can sound a bit odd in directions, so past the traffic light is often the most natural translation. Czech za covers this idea more broadly than English does.

Both are possible. Czech word order is much more flexible than English word order.

The version Za semaforem je knihovna vlevo puts the location za semaforem first, which sounds natural if you are giving directions step by step:

• Za semaforem = after the traffic light
• je knihovna vlevo = the library is on the left

A different order, such as Knihovna je vlevo za semaforem, is also grammatical, but it may sound like the speaker is focusing first on the library rather than the route.

So the word order often reflects what the speaker wants to emphasize, not a change in basic meaning.

Je is the 3rd person singular form of být (to be), so it means is.

• knihovna je vlevo = the library is on the left

In present-tense Czech, forms of to be are often omitted in some informal contexts, especially in headlines or notes, but in a normal full sentence like this, je is standard and expected.

So this sentence is a normal complete sentence:

• Za semaforem je knihovna vlevo.
• Past the traffic light, the library is on the left.

Because vlevo is a fixed adverb meaning to the left / on the left.

It works like:

• vlevo = on the left
• vpravo = on the right
• rovně = straight ahead

So you simply learn vlevo as a whole word.

Examples:

• Banka je vlevo. = The bank is on the left.
• Odbočte vlevo. = Turn left.

English speakers sometimes expect a preposition plus a noun, but Czech often uses a single adverb here.

Yes. Knihovna is the subject of the sentence, so it is in the nominative.

• knihovna = library
• je = is
• vlevo = on the left

So:

• knihovna je vlevo = the library is on the left

The subject stays in the nominative even though the sentence begins with a location phrase.

Not in normal English, and not really in Czech either. The sentence means that once you pass the traffic light, the library will be on your left.

A more natural English rendering is:

• Past the traffic light, the library is on the left.
• After the traffic light, the library is on the left.

So vlevo describes the library’s position relative to the person following directions.

Because Za semaforem is just a fronted adverbial phrase, not a separate clause.

Czech usually does not require a comma after a simple introductory phrase like this:

• Za rohem je pošta. = Around the corner there is a post office.
• Vpravo je banka. = On the right is a bank.

A comma would normally appear only if there were a more complex structure that required it.

A rough guide is:

za SE-ma-for-em ye KNI-hov-na VLE-vo

A few useful pronunciation notes:

• j in Czech sounds like English y in yes
  • so je sounds like ye
• ch is a separate sound in Czech, but it does not appear here
• h in knihovna is pronounced
• stress is usually on the first syllable of each word:
  • ZA
  • SEmaforem
  • JE
  • KNIhovna
  • VLEvo

Yes, that is grammatical. Czech allows this kind of variation.

However, the placement of vlevo can slightly change the feel of the sentence:

• Za semaforem je knihovna vlevo.
  A very natural directions-style sentence.
• Za semaforem je vlevo knihovna.
  Also possible, but can sound a bit more marked or focused.

In everyday direction-giving, the original version is very natural.

A good summary is:

1. za + instrumental for location

   • za semaforem = past / behind the traffic light

2. je is the present tense of být

   • je = is

3. knihovna is the subject, so it is nominative

   • knihovna = library

4. vlevo is an adverb

   • vlevo = on the left

5. Czech word order is flexible

   • starting with Za semaforem is natural when giving directions

So the sentence is a nice example of how Czech often builds directions: [location phrase] + [is] + [place] + [direction].