By itself it's no more left or right than the issue of prejudice is generally given that there are feminists, homosexuals and transgender people who are conservative politically but also woke in the original sense.

Very recently, in the last few years, it's been adopted as a pejorative term for far left identity politics. Now far left identity politics is a real thing, and it certainly is woke and probably deserves to have a pejorative term for it, but it has no ownership or exclusive claim on the term woke. unfortunately this may be a lost battle at this stage, but there are still a lot of people in the black community who have been using the term in its original meaning for generations and will doubtless continue to do so.