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This paper will be both to give an in-depth analysis as well as to present the new material on the notion of context-aware computing, an idea that computing can both sense and react accordantly based on its environment. The paper formalizes context-awareness process using coalgebraic language, including the coalgebraic definition of context-awareness, bisimulation between context-awarenesses, homomorphism between context-awarenesses and context-awarenesses as coalgebras. Discussions for further development based on this approach are also given.