Learn Hardcore Czech: V neděli je knihovna zavřená, proto si musím půjčit knihy už v sobotu. - On Sunday the library is closed, so I have to borrow books already on Saturday.

Knihovna je zavřená. – completely standard, colloquial, common speech.
Knihovna je zavřena. – also correct, a bit more formal / written / neutral.
In your sentence: je knihovna zavřená – same words, just slightly different word order.

So the important part is that zavřená agrees with knihovna (feminine), and the verb je (is) links them.

Why is the word order je knihovna zavřená and not knihovna je zavřená?

Czech word order is more flexible than English. Both:

V neděli je knihovna zavřená…
V neděli je zavřená knihovna…
V neděli knihovna je zavřená…

are grammatically possible.

In your sentence, the focus is on the state (closed) rather than on which place is closed. The pattern:

V neděli je knihovna zavřená…

feels neutral and natural in spoken Czech. If you really wanted to emphasize knihovna (as opposed to some other place), you might say:

V neděli je zavřená knihovna, ne pošta.

Why is the present tense je used for Sunday, which is in the future from now?

In Czech, scheduled or regular future events are very often expressed with the present tense, similar to English when you say:

The library is closed on Sunday. (even if “this Sunday” is in the future)

So V neděli je knihovna zavřená can mean:

As a general rule: On Sundays, the library is closed.
Or more contextually: This coming Sunday, the library is (going to be) closed.

If you wanted to make the future explicit, you could say:

V neděli bude knihovna zavřená. – The library will be closed on Sunday.

But the simple je is perfectly natural for schedules and known future states.

What is the difference between proto, takže, and protože here?

All three are related, but they behave differently:

proto – “therefore”, “for that reason” (an adverbial connector)
- V neděli je knihovna zavřená, proto si musím půjčit knihy už v sobotu.
  The library is closed on Sunday, therefore I must borrow the books already on Saturday.
It introduces a result.
takže – “so”, “so that / therefore” (more colloquial, like English “so”)
- V neděli je knihovna zavřená, takže si musím půjčit knihy už v sobotu.
Very common in speech; slightly less formal than proto.
protože – “because” (subordinating conjunction)
- Musím si půjčit knihy už v sobotu, protože je knihovna v neděli zavřená.
  I must borrow the books on Saturday because the library is closed on Sunday.

So:

proto / takže = introduce the consequence / result
protože = introduces the reason / cause

Why is the clitic si placed after proto: proto si musím půjčit, and not proto musím si půjčit?

Czech has clitics – short, unstressed words (like si, se, mi, ti, ho, by, jsem) that usually appear in the second position in a clause.

In the clause proto si musím půjčit knihy:

1st position: proto (therefore)
2nd position: si (clitic)
then: musím půjčit knihy

Placing si as proto musím si půjčit sounds unnatural; it breaks the usual “second position” rule for clitics.

So:

✅ proto si musím půjčit knihy
❌ proto musím si půjčit knihy (grammatical for some speakers in special emphatic contexts, but generally felt as odd or non‑standard)

What does si add to the verb půjčit? Why not just musím půjčit knihy?

Without si, půjčit usually means to lend (to someone):

Musím půjčit knihy kamarádovi. – I must lend the books to my friend.

With si, it becomes půjčit si – to borrow (for oneself):

Musím si půjčit knihy. – I must borrow books (for myself).

So in your sentence:

… proto si musím půjčit knihy už v sobotu.

means “I have to borrow the books (for myself)” rather than “I have to lend the books (to someone)”. The reflexive si indicates the action is for the subject’s own benefit.

What is the difference between půjčit si and půjčovat si?

This is about aspect: perfective vs. imperfective.

půjčit si – perfective, a single, completed act: to borrow (once, successfully)
- Musím si půjčit knihy. – I have to borrow the books (on one occasion).
půjčovat si – imperfective, repeated, ongoing, or in progress: to be borrowing, to borrow regularly
- Často si půjčuji knihy z knihovny. – I often borrow books from the library.

In your sentence, musím si půjčit refers to one specific act (go on Saturday and borrow them), so the perfective půjčit si is correct.

Why is it půjčit knihy and not půjčit si knihy immediately together? The word si is separated: si musím půjčit knihy.

The verb phrase is still půjčit si, but the clitic si has moved to the “second position” of the clause:

Full logical order: musím půjčit si knihy (easier to see both parts)
Actual natural order: musím si půjčit knihy or, with proto in front:
proto si musím půjčit knihy

So even though si is not right next to půjčit in the surface order, Czech speakers still understand it as part of the reflexive verb půjčit si. Clitics move for syntactic reasons, but their relationship with the verb stays the same.

Why is it knihy here, and what case is that?

The noun kniha (book) declines like this (singular vs. plural nominative/accusative):

Singular:
- Nominative: kniha (a/the book)
- Accusative: knihu (I read a book)
Plural:
- Nominative: knihy (books)
- Accusative: knihy (I read books)

After a verb like půjčit (si), the direct object is in the accusative case. Here it’s plural:

půjčit (si) knihy – to borrow books.

So knihy is accusative plural, which just happens to look the same as nominative plural.

Why is it v sobotu and not v sobota?

Exactly the same reason as with v neděli: you need the locative case after v when talking about “when” something happens.

The noun sobota (Saturday) declines (singular):

Nominative: sobota
Locative: sobotě (older / more formal), but in modern Czech the time expression is fixed as:
v sobotu (not v sobotě)

So the idiomatic time expression is:

v sobotu – on Saturday.

Similarly:

v pondělí – on Monday
v úterý – on Tuesday
ve středu – on Wednesday
ve čtvrtek – on Thursday
v pátek – on Friday
v sobotu – on Saturday
v neděli – on Sunday

All of them use forms that correspond to the locative.

What does už add in už v sobotu?

už often translates as already or by (then) and adds a nuance of earlier than expected or not later than that time.

Musím si půjčit knihy v sobotu. – I have to borrow the books on Saturday. (neutral)
Musím si půjčit knihy už v sobotu. – I have to borrow the books already on Saturday / as early as Saturday (because Sunday is too late).

So už v sobotu emphasizes that Saturday is the latest possible time to do it, or that you are doing it earlier than you would otherwise like.