  We give analogues of the Auslander correspondence for two classes of
triangulated categories satisfying certain finiteness conditions. The first
class is triangulated categories with additive generators and we consider their
endomorphism algebras as the Auslander algebras. For the second one, we
introduce the notion of $[1]$-additive generators and consider their graded
endormorphism algebras as the Auslander algebras. We give a homological
characterization of the Auslander algebras for each class. Along the way, we
also show that the algebraic triangle structures on the homotopy categories are
unique up to equivalence.
