The punishment being justified depends entirely on whether or not the law itself is just (e.g. capital punishment is largely considered to be a legal injustice)

The punishment being justified depends entirely on whether or not the law itself is just (e.g. capital punishment is largely considered to be a legal injustice). Whether or not responsibility is transitive a separate, but important, matter. As it turns out, responsibility for statutory laws is only transitive when applied to a party to the statute (e.g. a lawmaker or the enforcer). A thief is responsible for restitution because depriving someone of their property is violation of natural law. A party who breaks a contract is responsible for fulfilling the terms they agreed to, because they made that agreement. A thief, or a party who breaks a contract, is not responsible for a _statutory law_ enforced against them (unless they are a law maker or the law enforcer). The creators and enforcers of the statue are responsible for creating it and enforcing it. They are responsible for its outcome, whether that is a dead murderer or a dead baby.