Differential privacy (DP) is a widely applied paradigm for releasing data while maintaining user privacy. Its success is to a large part due to its composition property that guarantees privacy even in the case of multiple data releases. Consequently, composition has received a lot of attention from the research community: there exist several composition theorems for adversaries with different amounts of flexibility in their choice of mechanisms. But apart from mechanisms, the adversary can also choose the databases on which these mechanisms are invoked. The classic tool for analyzing the composition of DP mechanisms, the so-called composition experiment, neither allows for incorporating constraints on databases nor for different assumptions on the adversary's prior knowledge about database membership. We therefore propose a generalized composition experiment (GCE), which has this flexibility. We show that composition theorems that hold with respect to the classic composition experiment also hold with respect to the worst case of the GCE. This implies that existing composition theorems give a privacy guarantee for more cases than are explicitly covered by the classic composition experiment. Beyond these theoretical insights, we demonstrate two practical applications of the GCE: the first application is to give better privacy bounds in the presence of restrictions on the choice of databases; the second application is to reason about how the adversary's prior knowledge influences the privacy leakage. In this context, we show a connection between adversaries with an uninformative prior and subsampling, an important primitive in DP. To the best of our knowledge, this paper is the first to analyze the interplay between the databases in DP composition, and thereby gives both a better understanding of composition and practical tools for obtaining better composition bounds.