For example, the `number of functions’ rule (page X\pageref{p:num - fns}) will fail, and there will be further unpleasant surprises higher up the tower. The diagonal functor $ \Delta{\colon}\linebreak[0] \mathbf{Set} \to \mathbf{Set} \times \mathbf{Set} $ is defined by $ \Delta(A) = (A, A) $ for all sets $ A $ . Exhibit left and right adjoints to $ \Delta $ .