What is a possessive phrase (murakkab idāfī)?
Two nouns stand side by side and the second explains the first. The first is the مُضَاف (mudāf); the second is the مُضَاف إِلَيْه (mudāf ilayh).
Every idāfa has exactly two parts: the mudāf, which loses its tanwīn, and the mudāf ilayh, which is always majrūr.
In كِتَابُ مُحَمَّدٍ (kitābu muhammadin), “Muhammad’s book”, كِتَاب is the mudāf and مُحَمَّد is the mudāf ilayh. The mudāf loses its tanwīn, and the mudāf ilayh takes a kasra — it is always majrūr.
A hidden preposition is what the relation stands on. بَابُ خَشَبٍ (bābu khashabin) means a door of wood. صِيَامُ النَّهَارِ (siyāmu al-nahāri) means fasting in the day. كِتَابُ مُحَمَّدٍ means the book belonging to Muhammad.
Like most Arabic grammar terms, إِضَافَة (idāfa) was taken from everyday speech and then given one precise technical sense.
In everyday Arabic it means adding one thing to another.
أَضَافَ ضَيْفًا (adāfa dayfan) — “he took in a guest”
In Arabic grammar it means joining one noun to another so that the second defines the first.