While the Cobordism Hypothesis provides a translation between topological and categorical structures, the subject of fusion categories arising from representations of finite groups has shown the need for a robust theory of duality in monoidal 2-categories. This book intertwines 3-dimensional category theory with enriched and symmetric monoidal 2-categories to present a comprehensive foundation for duals in dimension two. This framework is presented using wire diagrams, a typographical tool that combines the geometric appeal of ordinary string diagrams with the clarity of standard commutative diagrams in category theory. Chapters on enriched 2-categories, closed 2-categories, and compact closed 2-categories build the theory from the ground up and in full detail. Shorter appendices on technical topics from higher category theory such as icons, strictification theorems, and computads provide readers with concise explanations of key results, as well as references for researchers interested in more depth. The book concludes with a chapter of examples from algebra and topology, including a detailed construction for the example of cobordisms.