As soon as possible, open the cupboard with the paper strips to detect sigils. You might get 1 or 2 during the first body embalment.
Once you have 2 sigils, you normally can already narrow it down to only a few possible Demons.
Continue with the story and the embalments until you get the 3rd sigil (normally during the second body). Most likely you will have a name now.
If you try to figure out which is the demon body to burn (and are correct) then you will finish this shift successfully and get "I Know You" achievement. If on top of that, you were also quick, then you can even get the speed run achievement.
IMPORTANT: Dont forget to place the Mark on the body before burning it!