Season Finale revelations...

So.... Just to backtrack... Red told Lizzy he would never lie to her- when Lizzy asked Red if he was her father, he paused and said no. Not because he isn't her biological father but because he believes he is no longer that same man. 

Red does not know the identity of his 'adversary'. Liz actually holds the key to that. 'Berlin' is Reds father-in-law, Berlin is Lizzys grandfather. Red is Lizzys father. Berlin falsely blames Red for his daughter (Lizzys mother) death and dismemberment at the hands of the stewmaker.