What is the possessor (muḍāf ilayh)?
Two isms stand side by side and the second tells you whose the first is, or what it is made of, or where it belongs. The first is the مُضَاف; the second is the مُضَاف إِلَيْه, and it is majrūr.
كِتَابُ زَيْدٍ (kitābu zaydin), Zayd's book. كِتَاب takes whatever i'rab its own role in the sentence needs; زَيْدٍ is majrūr because of the construction alone — no preposition is written.
Every iḍāfa has exactly two parts: the muḍāf, which loses its tanwīn, and the muḍāf ilayh, which is always majrūr.
That is the point of this second route to jarr: the preposition is understood, not spoken. The books say the relation carries an unwritten مِنْ, فِي or لِ — بَابُ خَشَبٍ, a door of wood; صَلَاةُ اللَّيْلِ, the prayer in the night; كِتَابُ زَيْدٍ, the book belonging to Zayd.
One rule follows immediately: the مُضَاف must be stripped of its tanwīn, and of anything standing in tanwīn's place — كِتَابُ زَيْدٍ, never كِتَابٌ زَيْدٍ; مُسْلِمِي مِصْرَ, not مُسْلِمُونَ مِصْرَ.
The two roads to jarr are not the same road.
After a حَرْف جَرّ the particle is written: فِي الْبَيْتِ.
In an إِضَافَة the particle is understood: كِتَابُ زَيْدٍ carries a silent لِ.
The visible difference is on the *first* word, not the second: a muḍāf loses its tanwīn. كِتَابٌ with tanwīn cannot be a muḍāf; كِتَابُ without it can be nothing else.