Synthetic data are becoming increasingly important for computational studies of cophylogeny, including machine learning, benchmarking, and method testing. Several generators have been proposed to produce such data, but each relies on different assumptions about host-symbiont coevolution. These assumptions are often implicit and rarely examined, even though results can depend strongly on the synthetic model being used. In this article, we present a systematic structural analysis of representative cophylogeny generators under controlled scenarios. The goal is to make their assumptions explicit and to understand how these choices shape the synthetic data they produce as well as the conclusions that may be drawn from them.