Magistra discipulos monet ut atramentum bene servent et ut codices diligenter teneant.

The sentence has:

• a main clause: Magistra discipulos monet
• two subordinate clauses introduced by ut:
  • ut atramentum bene servent
  • et ut codices diligenter teneant

So the pattern is:

The teacher warns/advises the students [to do X] and [to do Y].

Latin uses ut + subjunctive here where English often uses to or that.

Magistra is the subject of the main verb monet, so it must be in the nominative case.

• magistra = the female teacher
• nominative singular, first declension

It is the person doing the warning/advising.

Because discipulos is the direct object of monet.

The verb moneo, monere often means warn, advise, or remind, and the person being warned or advised is put in the accusative:

• magistra discipulos monet = the teacher warns/advises the students

So:

• discipuli would mean the students as a subject
• discipulos means the students as an object

After verbs like moneo, Latin commonly uses ut + subjunctive to express an indirect command or purpose-like clause.

So:

• monet ut servent = she warns/advises them to preserve
• monet ut teneant = she warns/advises them to hold/keep

English often uses to after verbs like advise:

• She advises the students to preserve the ink and keep the books carefully.

Latin does this differently, using a full clause with ut.

They are subjunctive because they are in ut-clauses depending on a verb of warning/advising: monet.

This is a standard use of the subjunctive in Latin: the indirect command.

Forms:

• servent = present subjunctive, 3rd person plural of servo, servare
• teneant = present subjunctive, 3rd person plural of teneo, tenere

They are plural because the understood subject is the students.

The subject is understood, not stated explicitly: it is discipulos in sense, meaning the students.

More precisely, in English we think of the students as the people doing the actions in the subordinate clauses. In Latin, the actual subject of those ut-clauses is simply omitted because it is obvious from context.

If Latin wanted to state it explicitly, it would use discipuli in the nominative inside the subordinate clause, not discipulos.

So the idea is:

• The teacher advises the students that they should preserve the ink and keep the books carefully.

Latin often repeats ut when joining two parallel subordinate clauses:

• ut atramentum bene servent
• et ut codices diligenter teneant

This makes the structure very clear.

Latin could sometimes omit the second ut, but repeating it is perfectly normal and often stylistically neat. It is like saying:

• that they preserve the ink well, and that they hold the books carefully

Both are direct objects of their verbs, so both are in the accusative.

• atramentum = accusative singular neuter
  object of servent
• codices = accusative plural masculine
  object of teneant

So:

• servent atramentum = preserve the ink
• teneant codices = hold/keep the books

Because this is the present subjunctive, not the indicative.

For a first-conjugation verb like servo, servare, the present subjunctive uses -e- instead of the usual indicative vowel:

• indicative: servant = they preserve
• subjunctive: servent = that they preserve / should preserve

Since the clause is introduced by ut after monet, the subjunctive is required.

Teneant is the present subjunctive, 3rd person plural, of teneo, tenere.

For second-conjugation verbs, the present subjunctive uses -ea-:

• indicative: tenent = they hold
• subjunctive: teneant = that they hold / should hold

Again, the subjunctive is required because this is an ut-clause after monet.

They are adverbs.

• bene modifies servent: preserve well
• diligenter modifies teneant: hold/keep carefully

A native English speaker should notice that Latin often uses adverbs in much the same way English does, but the position is more flexible.

The word order is fairly flexible, but it is meaningful in terms of emphasis and style.

Here we get:

• Magistra discipulos monet first: the main action
• then the two things the students are told to do

Within the subordinate clauses, the objects come before the verbs:

• atramentum ... servent
• codices ... teneant

That is very common in Latin. English depends more on word order; Latin depends more on case endings.

So even if the order changed, the grammatical relationships would still be clear because of forms like:

• discipulos = accusative
• codices = accusative
• magistra = nominative

Yes. Moneo can cover a range including:

• warn
• advise
• remind

Which English word sounds best depends on context.

In this sentence, if the teacher is giving careful instructions, advises may sound natural. If she is cautioning them to take care of the ink and books, warns may also fit.

So the grammar stays the same even if the exact English wording changes.