Is punishment an intrinsic part of justice? I would say no punishment but rather a consequence. If this then that
I would like to think that some of the core components of justice are consistency and equity. The same results for a given input.
The idea of justice extends far beyond our legal system into nature itself. The laws of physics, the laws of our universe are justice embodied. Gravity wirks the same whether you’re a “good” or “bad” person
In that respect I would say that judgement actually has no place in justice. Judgement is based entirely on the perspective of the one judging, even when they are trying to be impartial.
We try with our rules and laws to approximate the fundamental justice we see around us in nature