Harry did not say “James.” He said “Kill me like you killed him.” They were speaking of James, but “him” could also mean killing Dumbledore, the act that is tearing apart Snape’s soul, the act that turned him from being the traitor responsible for James and Lily’s deaths into an actual killer. Snape’s guilt over the deaths of Lily and James has distorted his perceptions of Harry every day for the past six years, and now, minutes after killing Dumbledore, Harry’s words slam him with a renewed realization of the enormity of that guilt. “DON’T—”
