  In this paper, we prove that there exists an equivalence between 2-category
of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the
1-category of stacky fans. For this purpose, we obtain two main results. The
first is to investigate a combinatorial aspect of the 2-category of toric
algebraic stacks defined in \cite{I2}. We establish an equivalence between the
2-category of toric algebraic stacks and the 1-category of stacky fans. The
second is to give a geometric characterization theorem for toric algebraic
stacks.
