DISTRICT ATTORNEY TALBOT: This will be the very definition of swift justice.
The District Attorney (Shaft!) could not be more wrong here. He says this when the greed murder is discovered, but there's a long road ahead. It's not the authorities that will be doling out justice; it's John Doe.
SOMERSET: He's preaching.
MILLS: He's punishing.
To John Doe, these two things seem to be one and the same. He's like Jonathan Edwards resurrected.
SOMERSET: But you got to be a hero. You want to be a champion, well, let me tell you, people don't want a champion.
Mills, unlike Somerset and many of the other cops, does want there to be justice. He thinks that the world can be a fair place where people who do evil are punished. Hmm—John Doe wants the same thing, doesn't he?
JOHN DOE: I won't deny my own personal desire to turn each sin against the sinner.
Just as Mills gets a thrill out of pursuing Doe, Doe gets a thrill out of committing his murders. Both of them are proud to be doling out their own version of justice.
JOHN DOE: Don't ask me to pity those people. I don't mourn them any more than I do the thousands that died at Sodom and Gomorrah.
The story of Sodom and Gomorrah is a biblical example of God dealing out a version of justice. Doe feels like God, and he feels the whole world has become Sodom and Gomorrah.
SOMERSET: David. If you kill him, he will win.
Somerset is telling Mills not to take justice into his own hands. It's not his to administer. But, like Doe, Mills does "become wrath" because he cannot suppress the urge to dole out justice for what Doe did to his wife.