Magistra bibliothecae praeest et vult ut omnes lectioni diligenter intersint.

Because praeest comes from praeesse, and praeesse takes the dative, not the accusative.

So in this sentence:

• magistra = the subject
• bibliothecae = dative, to / over the library in the sense of being in charge of the library

A good vocabulary habit is to learn these together:

• praeesse + dative = to be in charge of
• intersum + dative = to be present at / take part in

So bibliothecae is not a direct object. It is the noun required by the verb.

By form alone, yes: bibliothecae can be either:

• genitive singular = of the library
• dative singular = to/for the library

But in this sentence it must be dative, because praeest requires a dative.

So the grammar tells you how to interpret the form.

Praeest is the 3rd person singular present indicative active of praeesse.

Breakdown:

• prae- = a prefix meaning before or in front
• est = he/she/it is

So literally it is something like is set over, and idiomatically it means:

• is in charge of
• presides over
• is at the head of

Since the subject is magistra, praeest means she is in charge.

Because Latin often uses ut + subjunctive after verbs of wanting, especially when the subject of the second verb is different from the subject of the first verb.

Here:

• vult = she wants
• ut omnes ... intersint = that everyone be present / attend

So the structure is:

• magistra vult = the teacher wants
• ut omnes lectioni diligenter intersint = that everyone attend the lecture carefully

This is very common Latin syntax.

A useful comparison:

• vult venire = she wants to come
  same subject, so infinitive is natural
• vult ut omnes veniant = she wants everyone to come
  different subject, so ut + subjunctive

Because it is inside a clause introduced by ut after vult.

This kind of clause is often called a substantive clause of purpose or more simply a noun clause after a verb of wishing/commanding. The important practical point is:

• vult ut ... intersint

After vult in this construction, Latin uses the subjunctive.

Intersint is the present subjunctive, 3rd person plural of intersum.

The present subjunctive fits the sense here: the teacher wants this to happen now or in the future.

Because intersum also takes the dative.

So:

• lectioni interesse = to be present at the lecture/reading
• literally something like to be among/at the lecture

This is another verb that English speakers often find surprising, because English would usually use a direct object or a preposition, but Latin uses the dative.

So in this sentence:

• omnes lectioni intersint = everyone should attend the lecture

Here omnes is nominative plural and acts as the subject of intersint.

It means:

• all (people)
• everyone

Latin often uses adjectives by themselves, with the noun understood. So omnes here really means something like all people.

You can tell it is the subject because:

• intersint is 3rd person plural
• omnes is plural and matches that verb

Magistra is nominative singular and is the subject of both main verbs:

• praeest
• vult

So the structure is:

• Magistra ... praeest et vult ...
• The teacher / female leader is in charge ... and wants ...

Depending on context, magistra can mean female teacher, mistress, or female leader/head. Grammatically, though, its job here is simple: it is the subject.

Latin word order is much freer than English word order. Diligenter is an adverb meaning carefully, attentively, or diligently, and it modifies intersint.

Placing it before the verb is perfectly normal. Latin often puts adverbs near the word they modify, but it does not have to follow a fixed English-style order.

So these would all be possible Latin patterns in principle:

• diligenter intersint
• intersint diligenter
• sometimes even with other words in between

The chosen order may slightly affect emphasis, but the basic meaning stays the same.

The verb endings show you.

• vult is 3rd person singular = he/she/it wants
• intersint is 3rd person plural subjunctive = they should be present

Now match them with the nouns:

• magistra = singular, so it matches vult
• omnes = plural, so it matches intersint

Latin often relies on agreement like this more than English does.

Yes, and that is very helpful to notice.

• praeesse = prae- + esse
• intersum = inter- + sum

Many compounds of sum are irregular in forms but very common in meaning. They often keep forms related to sum, such as:

• est
• sunt
• sit
• sint

That is why you see:

• praeest
• intersint

Recognizing the sum element makes these easier to learn.

It has two main parts joined by et:

• Magistra bibliothecae praeest
• et vult ut omnes lectioni diligenter intersint

So the pattern is:

• main clause 1: subject + verb + dative complement
• main clause 2: same subject understood + verb + ut clause

More explicitly:

• Magistra = subject
• bibliothecae praeest = is in charge of the library
• et vult = and wants
• ut omnes lectioni diligenter intersint = that everyone attend the lecture carefully

This is a very typical Latin sentence pattern: a straightforward statement followed by a clause expressing what someone wants.

The most useful things to memorize are the verb patterns:

• praeesse + dative = to be in charge of
• intersum + dative = to be present at / attend
• velle ut + subjunctive when one person wants someone else to do something

If you remember those three patterns, much of the sentence becomes easy to recognize.