Feser and Bessette structure their argument for capital punishment carefully. Their fourth general premise is that some wrongdoers deserve death, but they spell out as a fifth and separate premise the notion that someone has the authority to inflict that death upon the wrongdoer:
Public authorities have the right, in principle, to inflict on wrongdoers the punishments they deserve.
As I noted in my original post on the death penalty, the fact that someone deserves death does not of itself imply that any human being has the authority to impose it. Accordingly, F&B devote an entire subsection of their argument to showing that the government in particular does have that authority.