Cornwall enters with Edmund, carrying the letter reporting news of the invasion from France (which Gloucester mentioned to Edmund in 3.3). Edmund waffles, feigning remorse at having betrayed his father.
Edmund's handing over Gloucester as a traitor is an ultimate violation of the father-child bond.
Granting Edmund the title of Earl of Gloucester, Cornwall then sends him to find his father and arrest him. Edmund assents, although he continues to lament a 'sore conflict' (3.5.23) between his duty and his blood. Cornwall reassures Edmund that he, Cornwall, will be Edmund's "dearer father" (26) from here on.
Cornwall suggests that just as he can take on the traditional legal role of a father in granting a political title that would usually come from inheritance, he can also replace a biological father of flesh and blood. Incidentally, as he falls for Edmund's performance of hesitation about turning in Gloucester, he misjudges Edmund just as Gloucester and Edgar have before.