Language is a product of both biological and cultural evolution. Clues to the origins of key structural properties of language can be found in the process of cultural transmission between learners. Recent experiments have shown that iterated learning by human participants in the laboratory transforms an initially unstructured artificial language into one containing regularities that make the system more learnable and stable over time. Here, we explore the process of iterated learning in more detail by demonstrating exactly how one type of structure--compositionality--emerges over the course of these experiments. We introduce a method to precisely quantify the increasing ability of a language to systematically encode associations between individual components of meanings and signals over time and we examine how the system as a whole evolves to avoid ambiguity in these associations and generate adaptive structure.