Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 3 / c / Solution

The Yoneda embedding preserves categorical products: for any $a$, maps $a\to c\times Y$ correspond naturally to pairs of maps $a\to c$ and $a\to Y$. Hence $H_{c\times Y}\cong H_c\times H_Y$. Using the exponential in a presheaf category, the Yoneda lemma and the exponential universal property in $\mathcal{C}$ givesEvery bijection is natural in $c$, since it is given by precomposition or the natural exponential adjunction. Thus these component bijections are a natural isomorphism of categorical presheaves: