One of the biggest mysteries in current science is how subjective experience, or consciousness, arises from objective substance and its physical interactions, such as human brains. Since 1990s, empirical and scientific studies on the relationship between consciousness and brain have advanced massively, especially thanks to neuroscientific approaches. Despite its empirical progress, there remains skeptical philosophers, cognitive scientists, and psychologists, who consider the science of consciousness is impossible,partly because the concept of consciousness is so difficult to define. Due to this difficulty, they argue, scholars who claim that they are empirically researching consciousness even do not know what they themselves are talking about. These skeptics hold that scientific methods cannot be applied to concepts that are not possible to define. In this article, we argue that consciousness is possible to rigorously define in a strict mathematical sense. To build this logic, we introduce category theory, which is a theory developed in mathematics in the latter half of the 20th century. Category theory is a framework originally invented to deal with relationships among objects, in particular between algebra and geometry. In recent years, category theory has been recognized for its potential to be applied to consciousness research. Throughout this paper, we propose several concrete examples of Consciousness Category and, eventually, we conclude that we can apply "Yoneda's lemma" to Consciousness Category. Yoneda's lemma, one of the most fundamental and powerful tools in category theory, says, in simple terms,that definitions of any concept is the same as descriptions of all relationships between the concept and the others. This striking viewpoint, which is founded mathematically,provides the validity to the act of defining consciousness through descriptions of relationships. We end with a future perspective; enriching Consciousness Category will provide a common language among researchers who disagree in some aspects of their respective definitions of consciousness. Common language is a necessary component for the big breakthrough to solve the mystery of consciousness.