This is the 10675th most frequent Polish word.
dostojnemu
"To the dignified" or "to the noble."
Here 'dostojnemu' describes someone esteemed or of high regard in the dative case, fitting the respectful attitude toward the guest.
Wszyscy z szacunkiem podeszli do dostojnemu gościowi.
Everyone approached the distinguished guest with respect.
The term 'dostojnemu' is used to denote reverence for the professor, showing esteem, again in the dative case.
Podczas ceremonii wręczono nagrodę dostojnemu profesorowi.
During the ceremony, the award was presented to the distinguished professor.
In this sentence, 'dostojnemu' is attributing respect and high status to the speaker, in the context of how the audience perceived them during the act of speaking.