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We show that the Kazhdan-Lusztig category KLk of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1) has vertex algebraic braided tensor supercategory structure and that its full subcategory Okfin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)