Numerous examples of duality appear throughout this book. Given categories $ \mathscr{A} $ and $ \mathscr{B} $ , there is a product category $ \mathscr{A} \times \mathscr{B} $ , in which $ \ob(\mathscr{A} \times \mathscr{B}) & = \ob(\mathscr{A}) \times \ob(\mathscr{B}), \ (\mathscr{A} \times \mathscr{B})((A, B), (A’, B’)) & = \mathscr{A}(A, A’) \times \mathscr{B}(B, B’). $ Put another way, an object of the product category $ \mathscr{A} \times \mathscr{B} $ is a pair $ (A, B) $ where $ A \in \mathscr{A} $ and $ B \in \mathscr{B} $ .