The Iterated Immediate Snapshot model (IIS), due to its elegant topological representation, has become standard for applying topological reasoning to distributed computing. In this paper, we focus on relations between IIS and the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS). We grasp equivalences between subsets of runs of AS and IIS (we call them sub-IIS and sub-AS models). To establish an equivalence between a sub-AS model M and a sub-IIS model M ′, we need two algorithms, a forward simulation F: AS → IIS and a backward simulation B: IIS → AS, such that B(F (M)) ⊆M and F (B(M ′)) ⊆M ′. AS and IIS are provided with such simulations and, thus, they have the same task computability power. However, the relations between proper sub-AS and sub-IIS models remained unclear until now. In this paper, we present a two-way simulation protocol that provides an equivalent sub-IIS model for any adversarial sub-AS model, i.e., for any sub-AS model specified by the sets of live processes. We achieve the result by ensuring that, under the two-way simulation, the set of live processes in an AS run coincides with the set of fast processes in the simulated IIS run, and vice versa. 1