In this chapter we finally give the formal definition of a category. We build up to each part of the definition first, also explaining that definitions of abstract mathematical structures often take the form of data, structure, and properties. We start with the data, which is objects and arrows. Then we give the structure, which is composition and identities. We discuss the fact that arrows can be drawn in many different ways while still giving the same abstract structure. We see that identities are a generalization of reflexivity, and composition is a generalization of transitivity. Finally, we give the properties or axioms, which are the unit laws and associativity of composition. We include a brief mention of size issues, and also describe how associativity can be encapsulated geometrically as a tetrahedron. We discuss the idea of drawing diagrams in category theory, in which we omit identities and composites for efficiency, and introduce the notion of commutative diagrams. The chapter concludes with a discussion of the importance of composition in a category.