Mi compañera tiene preparada la bibliografía, aunque todavía le falta comprobar una fecha.

Because compañera is the feminine form of compañero. It tells you the colleague/classmate/coworker is female.

Also notice that mi does not change for masculine vs feminine:

• mi compañero
• mi compañera

It only changes for number:

• mi compañera
• mis compañeras

This is a very common Spanish structure:

tener + direct object + past participle

Here:

• tiene = has
• preparada = prepared
• la bibliografía = the bibliography

It expresses a resulting state: the bibliography is already in a prepared state.

So tiene preparada la bibliografía is not just about the action of preparing it, but about the fact that it is now ready.

Because in this structure, the past participle agrees with the direct object.

The direct object is la bibliografía, which is:

• feminine
• singular

So the participle must also be:

• feminine
• singular

That gives preparada.

Compare:

• Tiene preparado el informe.
• Tiene preparada la bibliografía.
• Tiene preparados los apuntes.
• Tiene preparadas las fotocopias.

Yes, but it is not exactly the same.

• Ha preparado la bibliografía focuses more on the completed action.
• Tiene preparada la bibliografía focuses more on the current result/state: the bibliography is ready.

So both are possible in many contexts, but this sentence prefers tiene preparada because it contrasts nicely with the unfinished detail in the second clause: everything is basically ready, but one thing is still missing.

Spanish often uses the definite article where English might or might not.

Here la bibliografía refers to a specific bibliography already understood in context, such as the bibliography for a paper, project, or assignment.

So la sounds natural because speaker and listener both know which bibliography is being talked about.

Because aunque can be followed by either the indicative or the subjunctive, depending on meaning.

Here it introduces a real, known fact:

• she still has one date left to check

So Spanish uses the indicative:

• aunque todavía le falta...

Very roughly:

• aunque + indicative = although/even though, presenting something as factual
• aunque + subjunctive = even if, or although with uncertainty/hypothesis

Example contrast:

• Aunque está cansada, sigue trabajando. = She is tired; that is a fact.
• Aunque esté cansada, seguirá trabajando. = Even if she is tired...

Todavía means still or sometimes yet, depending on context.

In this sentence it shows that one task remains unfinished:

• todavía le falta comprobar una fecha

So it adds the idea of not finished yet.

Its position is flexible, but this placement is very natural:

• aunque todavía le falta...

You could also hear:

• aunque le falta todavía comprobar una fecha

But the original version sounds smoother and more standard here.

This is a very important pattern.

faltar + infinitive means:

• to still need to do something
• to have something left to do

So:

• le falta comprobar una fecha = she still needs to check one date

Literally, faltar works a bit like gustar:

• something is lacking to someone

So the structure is closer to:

• To her, checking one date is still lacking

That sounds odd in English, but it is natural in Spanish.

Le refers to mi compañera.

It is an indirect object pronoun, because with faltar, the person affected is treated as an indirect object:

• A mi compañera le falta...

Even when mi compañera is already mentioned, Spanish still uses the pronoun:

• Mi compañera... le falta...

That is normal.

It is le, not la, because la would be a direct object pronoun, and that is not how faltar works.

In the first clause, yes:

• Mi compañera tiene preparada la bibliografía

There, mi compañera is clearly the subject.

In the second clause, the meaning is still about her, but grammatically the structure changes:

• le falta comprobar una fecha

With faltar, the person is not the subject in the same way. The idea that is missing or left to do is the grammatical center of the expression, while le marks the person affected.

So semantically the sentence is still about your colleague, but the internal grammar of the second clause is different.

Because it means one date among the various details in the bibliography, not a uniquely identified date already singled out.

So una fecha presents it as:

• one date
• some date
• a remaining date that still needs checking

If the context had already identified a specific date, la fecha could be possible, but una fecha is the natural choice here.

Yes. A few natural alternatives are:

• Mi compañera ha preparado la bibliografía, aunque todavía le falta comprobar una fecha.
• Mi compañera tiene la bibliografía preparada, aunque todavía le falta comprobar una fecha.
• Mi compañera tiene preparada la bibliografía, pero todavía le queda comprobar una fecha.

These are all similar, but the original sentence is very natural and nicely balanced:

• first, it highlights what is already done
• then, it adds the small thing still pending

Yes, comprobar is very common in Spain for check or verify.

Depending on context, you might also hear:

• verificar
• revisar
• confirmar

But comprobar una fecha sounds very natural in Spain, especially when checking whether a date is correct.