Translation:Your teacher is old, Duo's is young.
Duo + -n- (buffer vowel) + -in- (possessive) + -ki. The -ki, as I understand it, allows Duo'nun to function as a noun. Without -ki, Duo'nun would require a noun. E.g., Duo'nun öğretmen. It is similar to how, in English, you cannot say "Duo's teacher is old, my is young", but rather, you have to say "mine is young". It's the same principle, except that where English only does that with pronouns, Turkish does that with regular possessive nouns as well.