It has traditionally been assumed that information systems describe physical reality. However, this assumption is becoming obsolete as digital infrastructures are increasingly part of real-world experiences. Digital infrastructures (ubiquitous and scalable information systems) no longer merely map physical reality representations onto digital objects but increasingly assume an active role in creating, shaping, and governing physical reality. We currently witness an "ontological reversal", where conceptual models and digital infrastructures change physical reality. Still, the fundamental assumption remains that physical reality is the only real world. However, to fully embrace the implications of the ontological reversal, conceptual modeling needs an "institutional turn" that abandons the idea that physical reality always takes priority. Institutional reality, which includes, for example, institutional entities such as organizations, contracts, and payment transactions, is not simply part of physical reality detached from digital infrastructures. Digital infrastructures are part of institutional reality. Accordingly, the research question we address is: What are the fundamental constructs in the design of digital infrastructures that constitute and transform institutional reality? In answering this question, we develop a foundation for conceptual modeling, which we illustrate by modeling the institution of open banking and its associated digital infrastructure. In the article, we identify digital institutional entities, digital agents regulated by software, and digital institutional actions as critical constructs for modeling digital infrastructures in institutional contexts. In so doing, we show how conceptual modeling can improve our understanding of the digital transformation of institutional reality and the prescriptive role of conceptual modeling. We also generate theoretical insights about the need for legitimacy and liability that advance the study and practice of digital infrastructure design and its consequences.