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In her essay, Lorde describes a consciousness that oppression can come from those within an oppressed group. For example, Lorde describes how she is oppressed by white women who, while oppressed themselves as women, oppress Lorde as a black person; Lorde is still oppressed as a woman by black men, and as a lesbian by fellow black women. This comes about because people within those groups might focus only on their particular oppressions and forget their relative privilege compared to some members of the same group. Additionally, those further oppressed are asked to ignore those differences in the name of unity to the one group. This demonstrates how power can be negatively utilized even in an oppressed situation if one forgets what relative privileges they may have.
Gloria Anzaldua proposes that people oppressed by multiple systems develop a 'boarderland' or 'hybrid' consciousness. It is impossible to separate any one part a person oppressed in such a way; that person is necessarily a combination of all parts involved. This is due to the fact that those people are forced to constantly go between the groups with which they are oppressed where they are coerced to becoming only that one aspect of themselves.
Both of these authors hammer in the point that, when oppressed, an individual cannot be asked to become only one portion of themselves. They are still a combination of everything that they are, and the specific oppression they face is a result of all those factors together. Anzaldua, however, is arguably more effective in getting this point across not only by actually naming and describing this idea as a new consciousness, but also by adding in the aspect of being a third world person.