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We show that pretriangulated dg categories enjoy a universal property and deduce that the passage to an orbit quotient commutes with the dg quotient. In particular, for a triangulated category with dg enhancement and an endofunctor, there exists a unique triangulated orbit category. As an application, we prove that for any connective, smooth and proper dg algebra A, its perfect derived category is equivalent to the generalized (X-1) -cluster category of A. This implies that the orbit m-cluster category of A is equivalent to the generalized m-cluster category of A, which implies a conjecture by Ikeda-Qiu for the case when A is a smooth proper graded gentle algebra.
Li et al. (Tue,) studied this question.