This paper mainly studies the formal concept analysis in an intuitionistic fuzzy formal context. The primitive notions in concept lattice theory are extended to the intuitionistic fuzzy environment. According to this view, the notion of crisp-intuitionistic fuzzy concept is proposed which originates from an defined extent operator and an intent operator. Some basic properties are discussed. As a result, it is proved that the family of all crisp-intuitionistic fuzzy concepts induced by this two operators just forms a complete lattice.