I agree with you, but here, we are talking about the legal process by which the person is given the death penalty. That can be unjust even if the death penalty itself is justified when used in a just manner.

Let's take it to first principles. If a guy walks up to you and tells you "I am going to kill you if you don't kill me first" do you have a right to kill him? I'd say yes, what do you think?

Now, you and a group of people are hanging out. Some guy walks up to you guys and said "I'm going to kill one of you, I promise." Do you as a group have the right to kill him? Again, I'd say yes.

We can take that and do this thought experiment and find where it stops being justified. The man says "I'm going to rape all your kids and I won't stop until I'm dead", he doesn't say it verbally but demonstrates it with his behavior, etc. Then, what qualifies as a group? Does a jury that is part of the group qualify? You'll find plenty of scenarios where it is not done in a just manner, or when what someone is going to do if left to their own behavior doesn't justify killing them, but you'll find plenty of scenarios where it is justified. Therefore, the death penalty is just, even if the way it is done in our society is unjust.