What is a conjunctive phrase (murakkab 'atfī)?

Two words are linked by a particle, and whatever ending the first takes, the second takes too. The particle is the حَرْف عَطْف (harf 'atf); the word after it is the مَعْطُوف (ma'tūf); the word before it is the مَعْطُوف عَلَيْه (ma'tūf 'alayh).

The ma'tūf 'alayh comes first, the harf 'atf stands between, and the ma'tūf follows — taking the same i'rab as the first.

In حَضَرَ مُحَمَّدٌ وَعَلِيٌّ (hadara muhammadun wa-'aliyyun), “Muhammad and Ali came”, مُحَمَّد is marfū' as the doer, and عَلِيّ is marfū' too — the wāw passed the ending on.

Nine particles do this work: الْوَاو، الْفَاء، ثُمَّ، أَوْ، أَمْ، بَلْ، لَا، لَٰكِنْ، حَتَّى. Each links in its own way — the wāw simply joins, while the fā' adds order and immediacy.

Like most Arabic grammar terms, عَطْف ('atf) was taken from everyday speech and then given one precise technical sense.

In everyday Arabic it means bending or inclining one thing toward another.

عَطَفَ عَلَيْهِ ('atafa 'alayhi) — “he was kind to him”

In Arabic grammar it means joining a second word to a first with a linking particle.