Paul lui fait un compliment sincère sur son potager, et la voisine lui sourit enfin.

Because faire un compliment à quelqu’un takes an indirect object in French.

• Paul fait un compliment à la voisine = Paul gives the neighbor a compliment.
• When à la voisine is replaced by a pronoun, it becomes lui:
  • Paul lui fait un compliment

Here, lui means to him or to her, depending on context. In this sentence, it means to her.

For the same basic reason: sourire also takes an indirect object in French.

• sourire à quelqu’un = to smile at someone
• La voisine sourit à Paul → La voisine lui sourit

So even though English says smiles at him, French uses the indirect object pronoun lui.

This is something English speakers often have to get used to: some French verbs that feel direct in English are indirect in French.

French often uses the expression faire un compliment à quelqu’un, which literally means to make/give someone a compliment.

There is also a verb complimenter, but faire un compliment is extremely common and very natural in everyday French.

So:

• Paul lui fait un compliment = very common, idiomatic
• Paul la complimente = also possible, but a bit different in feel

English often uses a single verb where French prefers a verb + noun expression.

Because adjective placement in French is different from English.

Most French adjectives come after the noun, and sincère is one of them:

• un compliment sincère

Some adjectives commonly come before the noun, especially short, frequent ones related to beauty, age, number, goodness, and size. But sincère normally comes after.

So the word order here is completely standard.

Here, sur means about or regarding.

So:

• un compliment sur son potager = a compliment about his/her vegetable garden

French often uses sur in this kind of context to mean on the subject of.

Other phrases are possible in French, but sur is very natural here.

By itself, son can mean his, her, or sometimes its, depending on the owner.

So grammatically, son potager does not tell you by itself whether the garden belongs to Paul or to the neighbor. You have to rely on context.

That is because French possessive adjectives agree with the thing possessed, not with the owner:

• potager is masculine singular
• so French uses son

If the possessed noun were feminine, you might still see son before it if the word begins with a vowel sound, but that is a separate pronunciation issue.

Because son/sa/ses depend on the gender and number of the noun being owned, not on the gender of the owner.

Here:

• potager is masculine singular
• so French uses son

Compare:

• son potager = his/her vegetable garden
• sa maison = his/her house
• ses fleurs = his/her flowers

So son does not mean the owner is male. It only matches potager.

French can certainly use elle, but repeating the noun is often done for clarity, especially when there are two people in the sentence.

Here, both Paul and the neighbor are involved, and there are already two instances of lui. Saying la voisine makes it immediately clear who is smiling.

Compare:

• Paul lui fait un compliment..., et la voisine lui sourit enfin.
  Very clear.
• Paul lui fait un compliment..., et elle lui sourit enfin.
  Also possible, but slightly less explicit.

So this is a stylistic choice that helps avoid ambiguity.

Enfin here means finally.

It suggests that the neighbor did not smile before, and that this smile happens after some delay, hesitation, or previous tension.

Positionally, French often places adverbs like enfin after the verb or near the end of the clause:

• la voisine lui sourit enfin

That sounds very natural.

Because the idea of at/to him is already built into lui.

In English, we say:

• The neighbor smiles at him

In French, the structure is:

• sourire à quelqu’un
• la voisine lui sourit

So lui already means to him / to her, and no extra word like at is needed.

Not exactly.

Un potager is specifically a vegetable garden or kitchen garden. It is more specific than the general English word garden.

French distinguishes between different kinds of gardens, for example:

• un jardin = a garden in general
• un potager = a vegetable garden

So potager tells you what kind of garden it is.

They are in the present indicative, third-person singular:

• Paul fait
• la voisine sourit

French often uses the present in examples, narratives, and general scene-setting, much like English can.

The forms come from:

• faire → il/elle fait
• sourire → il/elle sourit

Both subjects are singular, so the verbs are singular too.

Yes. For example:

• Paul fait un compliment sincère à la voisine sur son potager, et la voisine sourit enfin à Paul.

That version is grammatically correct, but it sounds heavier because the nouns are repeated. French often replaces those repeated à + person phrases with lui to sound smoother:

• Paul lui fait un compliment sincère sur son potager, et la voisine lui sourit enfin.

So the pronouns make the sentence more natural and less repetitive.