Position of Double Pronouns in French - Direct Object Pronoun + Indirect Object Pronoun

In French, if there are two objects pronouns in one sentence, they each take a specific position.

Position of double object pronouns in a sentence in French

These sentences use two pronouns:

Me/Te/Nous/Vous (Me/You/Us)

Je vous la donne.I'm giving it to you.

Tu me l'as demandé.You asked me it.

Vous nous les offrez.You offer them to us.

Lui/Leur (Her/Him/Them)

Il les leur a apportés.He brought them to them.

Tu les lui vends.You sell them to her.

Nous les leur avons donnés.We gave them to them.

Je la lui ai écrite I wrote it to her.

There are two important patterns to notice in these sentences that are different to English.  

1) the two pronouns both go before the conjugated verb (or auxiliary in compound tenses):
Je vous donne cette carte. -> Je vous la donne.
I'm giving you this card. -> I'm giving it to you.
2) the order is not the same when using lui/leur as it is for me/te/nous/vous.
Je la lui donne vs. Je te la donne
I give it to him/her   I give it to you.
The order is ALWAYS:
me/te/nous/vous   (before)  le/la/les/l' (before) lui/leur  

Ambiguous cases

Look at the following sentence:
Il le lui montre.
Here both the direct object pronoun (le) and the indirect object pronoun (lui) refer to a masculine singular noun.
Now, in English, these pronouns would be "it" if the noun was an inanimate object, and "him" if it's a person. However, in French, this distinction doesn't exist, which means that the sentence "Il le lui montre." could mean either He shows it to him or He shows him to it, depending on wider context:
Il montre son nouveau portable à mon frère. Il le lui montre.He shows his new mobile [US:cell phone] to my brother. He shows it to him.

Il montre le Prince au miroir magique. Il le lui montre.He shows the Prince to the magic mirror. He shows him to it.

The same ambiguity exists with two feminine singular nouns, or two plural nouns:
Nous la lui donnons.We're giving it to her.
We're giving her to it.

Je les leur montre.I'm showing them to them.

 
To avoid that ambiguity, you can choose to use only one of the two pronouns, and keep one of the referrents.
 
 
See also how to use direct and indirect pronouns:

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Examples and resources

Nous les leur avons donnés.We gave them to them.
Tu les lui vends.You sell them to her.
Il les leur a apportés.He brought them to them.
Je vous la donne.I'm giving it to you.
Il montre le Prince au miroir magique. Il le lui montre.He shows the Prince to the magic mirror. He shows him to it.
Nous la lui donnons.We're giving it to her.
We're giving her to it.
Je les leur montre.I'm showing them to them.
Il montre son nouveau portable à mon frère. Il le lui montre.He shows his new mobile [US:cell phone] to my brother. He shows it to him.
Je la lui ai écrite I wrote it to her.
Vous nous les offrez.You offer them to us.
Tu me l'as demandé.You asked me it.
Let me take a look at that...