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In this paper, category GIFS of generalized intuitionistic fuzzy sets(GIF) is built up. Topoi properties of category GIFS are studied. Firstly, it is proved that the category GIFS has all topoi properties except that it has no subobject classifiers, Secondly, it is proved that the category GIFS has middle object and consequently GIFS is a weak topos.Thirdly, by the use of theory of weak topos GIFS, the power object of an object in GIFS is studied.