A data structure describes a way to organize and store a collection of data. It defines primitive efficient functions and operations that can be applied to the data such as constructors, modifications and access maps on the data. In this work, we consider data structures on strings as combinatorial descriptions of structured words having a theory of normal forms defined by insertion algorithms. We show that an insertion map of a string data structure induces a product on data and we give necessary conditions making this product associative. Our construction allows us to give a rewriting description of the cross-section property for the structure monoid of a string data structure. We show how to compute a coherent presentation of the structure monoid made of rewriting rules defined by insertion on words and whose syzygies are defined as relations among the insertion algorithms. As an illustration, we show how our constructions can be applied to Chinese monoids by making explicit the shape of syzygies of the Chinese congruence.