In my opinion Didact's memories and personality are in Master Chief's head now, after the evolving process that the Librarian makes him...
Because after that passage happens that MC hears the voice of the Didact and Cortana does not! And this passage of memories is typical of Forerunners technology and civilization... :D
The real didact is dead but he is still alive in MC's head, as it happend with Bornstellar!