ظَبْيٌ = gazelle; ظَبْيِي = my gazelle.
The first yāʾ belongs to the word ẓaby itself; the second is the possessive yāʾ.
In رَأَيْتُ ظَبْيِي, ẓabyī is an accusative object with an estimated ending, and the attached possessive yāʾ is mudāf ilayh.
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Ẓabyī: Keep the Two Yāʾs Apart
Iḍāfah to Yāʾ al-Mutakallim and the Six Nouns
ظَبْيٌ = gazelle; ظَبْيِي = my gazelle.
The first yāʾ belongs to the word ẓaby itself; the second is the possessive yāʾ.
In رَأَيْتُ ظَبْيِي, ẓabyī is an accusative object with an estimated ending, and the attached possessive yāʾ is mudāf ilayh.