Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In
proof theory, proofs and
theorems are also mathematical objects.
Hersh, Reuben, 1997. What is Mathematics, Really? Oxford University Press.
Sfard, A., 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P., et al., Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design. Lawrence Erlbaum.
Stewart Shapiro, 2000. Thinking about mathematics: The philosophy of mathematics. Oxford University Press.