Learn Turkish: Rektör, enstitünün arşiv odasına girdi. - The rector entered the institute's archive room.

This is a chain noun compound made of three nouns: enstitü, arşiv, and oda. First, arşiv and oda form the indefinite compound arşiv odası (archive room). Then, we link enstitü to it using the genitive suffix -nün (institute's). The whole chain becomes enstitünün arşiv odası.

Why is there an extra -n- in odasına?

The word oda (room) first takes the compound marker -sı to become odası (part of arşiv odası). When we want to add the dative case suffix -a (meaning to or into), we need a buffer consonant. After a possessive or compound marker, Turkish always uses the letter -n- as a buffer, making it odasına.

Why does odasına take the dative suffix -a here?

The verb girmek (to enter) requires its destination to be in the dative case. In English, we just enter a room, but in Turkish, you grammatically enter into a room. That is why the compound ends with the dative -a instead of being a direct object.

Why does enstitünün have -nün instead of -nin?