Mater dicit se filiis adesse, sive laeti sint sive tristes.

There is no separate word for that here.

Latin very often expresses reported statements with an accusative + infinitive construction instead of a clause introduced by that. So:

• Mater dicit = Mother says
• se filiis adesse = that she is there for her children

So the idea of English that is built into the grammar rather than spelled out with a separate word.

Se is the reflexive pronoun, and it refers back to the subject of the main verb, which is mater.

So:

• Mater dicit se... = The mother says that she...
• se points back to mater

If Latin used eam, that would normally mean her referring to some other female, not back to the mother who is doing the speaking.

This is a very common pattern in indirect statement:

• main subject + verb of saying/thinking + se
  • infinitive

In Latin indirect statement, the subject of the infinitive is put in the accusative case.

So in:

• Mater dicit se filiis adesse

the verb dicit introduces an indirect statement, and inside that indirect statement:

• se is the subject
• adesse is the infinitive

Even though se is logically the subject of adesse, grammatically it appears in the accusative because that is how this construction works in Latin.

Because it is part of the indirect statement after dicit.

Latin commonly uses:

• a finite verb of saying, thinking, knowing, hearing, etc.
• followed by an accusative subject
• plus an infinitive

So here:

• dicit = says
• se = accusative subject of the indirect statement
• adesse = infinitive

That is the normal Latin way to say says that she is present / is there.

Adesse is the infinitive of adsum, adesse, adfui, which is a compound of sum with ad-.

It literally means something like:

• to be present
• to be at hand
• and in many contexts, to be there for, to help, or to support

So here it is not just physical presence. It can suggest being available or supportive toward the children.

Because adsum takes the dative of the person to whom someone is present or available.

So:

• adesse filiis = to be present to the children / to be there for the children

This is different from English, where we might use for. Latin does not need a preposition here; it uses the dative case instead.

Sive ... sive ... means whether ... or ....

So:

• sive laeti sint sive tristes = whether they are happy or sad

It sets up two alternatives and says that the main statement is true in either case.

You can think of it as a correlative pair:

• sive X sive Y
• whether X or Y

Sint is subjunctive, while sunt would be indicative.

After sive ... sive ..., Latin often uses the subjunctive when giving an alternative like whether ... or ..., especially in a more general or less concrete way. Since this clause is also within reported speech after dicit, the subjunctive is especially natural.

So:

• sive laeti sint sive tristes = whether they be happy or sad / more naturally whether they are happy or sad

In idiomatic English, we usually just translate with the normal present tense.

Because they are predicate adjectives with the understood subject of sint.

The idea is:

• filii laeti sunt = the children are happy
• filii tristes sunt = the children are sad

In the actual sentence, filii is not repeated, but it is understood from filiis and the context. So the adjectives are in the nominative plural to agree with that understood subject.

• laeti = nominative masculine plural
• tristes = nominative plural here as well

They are not agreeing with se, and they are not in the dative like filiis, because they belong with the subject of sint.

Because the verb is sint, which is plural, and the adjectives are plural too:

• laeti
• tristes
• sint

But mater is singular. So the mother cannot be the subject of sint.

The plural subject must be the children, understood from filiis and the overall sense:

• whether the children are happy or sad

Meanwhile, se still refers back to the singular mother as the subject of adesse.

Literally, filiis is from filius, which most directly means sons.

However, depending on context, translators may sometimes use children if that better fits the sense in English. Strictly speaking, if Latin wanted to say children in a way that clearly included both sons and daughters, it might use a different word, such as liberi.

So:

• literal value: to/for the sons
• possible idiomatic translation in context: to/for the children

The word order is natural, but Latin word order is much freer than English word order.

This sentence is arranged as:

• Mater dicit — main clause
• se filiis adesse — indirect statement
• sive laeti sint sive tristes — alternative clause

A more mechanical rearrangement would still mean the same thing, because the endings show the grammatical relationships. For example, Latin can place filiis before adesse without causing confusion, since filiis is clearly dative and adesse clearly belongs to the indirect statement.

So the order helps the flow, but the case endings do most of the grammatical work.