Category theory manages mathematical concepts from the highest point of view. It considers not the intrinsic structure of each object, but the relation with other objects. We usually first define a mathematical object by its components such as elements, then make morphisms with other objects conserving component-level properties. However, category theory suggests that initially setting all morphisms between objects also determines a property that is similar to those derived from componential definitions of objects. In this book, our main subject is objects, morphisms, and commuting diagrams.